Theorem clwwlknonex2lem2 16433

Description: Lemma 2 for clwwlknonex2 16434: 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 10525	. . . . . . . . . . . . . . . 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 11228	. . . . . . . . . . . . . . . . . 18 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
6		elfzo0 10520	. . . . . . . . . . . . . . . . . . 19 |- ( i e. ( 0..^ ( (♯ ` W ) - 1 ) ) <-> ( i e. NN0 /\ ( (♯ ` W ) - 1 ) e. NN /\ i < ( (♯ ` W ) - 1 ) ) )
7		nn0re 9505	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i e. NN0 -> i e. RR )
8	7	adantr 276	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> i e. RR )
9		nn0re 9505	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( (♯ ` W ) e. NN0 -> (♯ ` W ) e. RR )
10		peano2rem 8540	. . . . . . . . . . . . . . . . . . . . . . . . 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 9206	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) - 1 ) < (♯ ` W ) )
16	15	adantl 277	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> ( (♯ ` W ) - 1 ) < (♯ ` W ) )
17		lttr 8347	. . . . . . . . . . . . . . . . . . . . . . 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 11335	. . . . . . . . . . . . . 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 2238	. . . . . . . . . . . . 13 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> ( W ` i ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) )
30		peano2nn0 9536	. . . . . . . . . . . . . . . 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 8289	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> 1 e. RR )
33	8, 32, 13	ltaddsubd 8819	. . . . . . . . . . . . . . . . . . . . 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 11335	. . . . . . . . . . . . . 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 2238	. . . . . . . . . . . . 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 3776	. . . . . . . . . . . 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 2301	. . . . . . . . . . 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 2538	. . . . . . . . . 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 6057	. . . . . . . . . . . . . . . . . . . 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 9865	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
59		2cnd 9310	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC )
60		1cnd 8290	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> 1 e. CC )
61	58, 59, 60	subsub4d 8615	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - ( 2 + 1 ) ) )
62		2p1e3 9371	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 2 + 1 ) = 3
63	62	a1i 9	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. ( ZZ>= ` 3 ) -> ( 2 + 1 ) = 3 )
64	63	oveq2d 6066	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> ( N - ( 2 + 1 ) ) = ( N - 3 ) )
65		uznn0sub 9886	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 3 ) e. NN0 )
66	64, 65	eqeltrd 2309	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. ( ZZ>= ` 3 ) -> ( N - ( 2 + 1 ) ) e. NN0 )
67	61, 66	eqeltrd 2309	. . . . . . . . . . . . . . . . . . . 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 2309	. . . . . . . . . . . . . . . . . 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 9206	. . . . . . . . . . . . . . . . . . . . . 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 11335	. . . . . . . . . . . . . . 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 9506	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (♯ ` W ) e. NN0 -> (♯ ` W ) e. CC )
87		ax-1cn 8220	. . . . . . . . . . . . . . . . . . . . . 22 |- 1 e. CC
88		npcan 8482	. . . . . . . . . . . . . . . . . . . . . 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 5674	. . . . . . . . . . . . . . . . 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 2232	. . . . . . . . . . . . . . . . . . . 20 |- (♯ ` W ) = (♯ ` W )
95		ccatw2s1p1g 11333	. . . . . . . . . . . . . . . . . . . 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 2265	. . . . . . . . . . . . . . . 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 3776	. . . . . . . . . . . . . 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 11271	. . . . . . . . . . . . . . . . . . . . . . 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 3776	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( W ` 0 ) = X /\ W e. Word V ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` ( (♯ ` W ) - 1 ) ) , X } )
106	105	eleq1d 2301	. . . . . . . . . . . . . . . . . . . 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 2309	. . . . . . . . . . . . 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 11334	. . . . . . . . . . . . . 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 3776	. . . . . . . . . . . 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 2309	. . 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 9698	. . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) e. ZZ )
138		1zzd 9604	. . . . . . . . 9 |- ( W e. Word V -> 1 e. ZZ )
139	137, 138	zsubcld 9705	. . . . . . . 8 |- ( W e. Word V -> ( (♯ ` W ) - 1 ) e. ZZ )
140		fveq2 5670	. . . . . . . . . . 11 |- ( i = ( (♯ ` W ) - 1 ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) )
141		fvoveq1 6073	. . . . . . . . . . 11 |- ( i = ( (♯ ` W ) - 1 ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) )
142	140, 141	preq12d 3776	. . . . . . . . . 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 2301	. . . . . . . . 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 5670	. . . . . . . . . . 11 |- ( i = (♯ ` W ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) )
145		fvoveq1 6073	. . . . . . . . . . 11 |- ( i = (♯ ` W ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) )
146	144, 145	preq12d 3776	. . . . . . . . . 10 |- ( i = (♯ ` W ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } = { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } )
147	146	eleq1d 2301	. . . . . . . . 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 3740	. . . . . . . 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 3400	. 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 2203   A.wral 2520   u. cun 3209   {cpr 3690   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   < clt 8308   - cmin 8444   NNcn 9237   2c2 9288   3c3 9289   NN0cn0 9496   ZZcz 9577   ZZ>=cuz 9853  ..^cfzo 10476  ♯chash 11138  Word cword 11224  lastSclsw 11269   ++ cconcat 11278   <"cs1 11303  Vtxcvtx 16007  Edgcedg 16052

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-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

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-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-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

This theorem is referenced by:  clwwlknonex2  16434