Theorem wwlksnextwrd 26792

Description: Lemma for wwlksnextbij 26797. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 18-Apr-2021.)

Hypotheses

Ref	Expression
wwlksnextbij0.v	|- V = (Vtx ` G )
wwlksnextbij0.e	|- E = (Edg ` G )
wwlksnextbij0.d	|- D = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) }

Assertion

Ref	Expression
wwlksnextwrd	|- ( W e. ( N WWalksN G ) -> D = { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } )

Distinct variable groups:   w, G   w, N   w, W

Allowed substitution hints:   D( w)   E( w)   V( w)

Proof of Theorem wwlksnextwrd

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnextbij0.d	. 2 |- D = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) }
2		3anass 1042	. . . . 5 |- ( ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) <-> ( ( # ` w ) = ( N + 2 ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) )
3	2	bianass 842	. . . 4 |- ( ( w e. Word V /\ ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) <-> ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) )
4		wwlksnextbij0.v	. . . . . . . . . . 11 |- V = (Vtx ` G )
5	4	wwlknbp 26733	. . . . . . . . . 10 |- ( W e. ( N WWalksN G ) -> ( G e. _V /\ N e. NN0 /\ W e. Word V ) )
6		simpl 473	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) ) -> N e. NN0 )
7		simpl 473	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) -> w e. Word V )
8		nn0re 11301	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> N e. RR )
9		2re 11090	. . . . . . . . . . . . . . . . . . . . 21 |- 2 e. RR
10	9	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> 2 e. RR )
11		nn0ge0 11318	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> 0 <_ N )
12		2pos 11112	. . . . . . . . . . . . . . . . . . . . 21 |- 0 < 2
13	12	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> 0 < 2 )
14	8, 10, 11, 13	addgegt0d 10601	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN0 -> 0 < ( N + 2 ) )
15	14	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> 0 < ( N + 2 ) )
16		breq2 4657	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` w ) = ( N + 2 ) -> ( 0 < ( # ` w ) <-> 0 < ( N + 2 ) ) )
17	16	ad2antll 765	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> ( 0 < ( # ` w ) <-> 0 < ( N + 2 ) ) )
18	15, 17	mpbird 247	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> 0 < ( # ` w ) )
19		hashgt0n0 13156	. . . . . . . . . . . . . . . . 17 |- ( ( w e. Word V /\ 0 < ( # ` w ) ) -> w =/= (/) )
20	7, 18, 19	syl2an2 875	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> w =/= (/) )
21		lswcl 13355	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word V /\ w =/= (/) ) -> ( lastS ` w ) e. V )
22	7, 20, 21	syl2an2 875	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> ( lastS ` w ) e. V )
23	22	adantrr 753	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) ) -> ( lastS ` w ) e. V )
24		swrdcl 13419	. . . . . . . . . . . . . . . . . . . . 21 |- ( w e. Word V -> ( w substr <. 0 , ( N + 1 ) >. ) e. Word V )
25		eleq1 2689	. . . . . . . . . . . . . . . . . . . . 21 |- ( W = ( w substr <. 0 , ( N + 1 ) >. ) -> ( W e. Word V <-> ( w substr <. 0 , ( N + 1 ) >. ) e. Word V ) )
26	24, 25	syl5ibr 236	. . . . . . . . . . . . . . . . . . . 20 |- ( W = ( w substr <. 0 , ( N + 1 ) >. ) -> ( w e. Word V -> W e. Word V ) )
27	26	eqcoms 2630	. . . . . . . . . . . . . . . . . . 19 |- ( ( w substr <. 0 , ( N + 1 ) >. ) = W -> ( w e. Word V -> W e. Word V ) )
28	27	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) -> ( w e. Word V -> W e. Word V ) )
29	28	com12 32	. . . . . . . . . . . . . . . . 17 |- ( w e. Word V -> ( ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) -> W e. Word V ) )
30	29	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) -> ( ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) -> W e. Word V ) )
31	30	imp 445	. . . . . . . . . . . . . . 15 |- ( ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) -> W e. Word V )
32	31	adantl 482	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) ) -> W e. Word V )
33		oveq1 6657	. . . . . . . . . . . . . . . . . 18 |- ( W = ( w substr <. 0 , ( N + 1 ) >. ) -> ( W ++ <" ( lastS ` w ) "> ) = ( ( w substr <. 0 , ( N + 1 ) >. ) ++ <" ( lastS ` w ) "> ) )
34	33	eqcoms 2630	. . . . . . . . . . . . . . . . 17 |- ( ( w substr <. 0 , ( N + 1 ) >. ) = W -> ( W ++ <" ( lastS ` w ) "> ) = ( ( w substr <. 0 , ( N + 1 ) >. ) ++ <" ( lastS ` w ) "> ) )
35	34	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) -> ( W ++ <" ( lastS ` w ) "> ) = ( ( w substr <. 0 , ( N + 1 ) >. ) ++ <" ( lastS ` w ) "> ) )
36	35	ad2antll 765	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) ) -> ( W ++ <" ( lastS ` w ) "> ) = ( ( w substr <. 0 , ( N + 1 ) >. ) ++ <" ( lastS ` w ) "> ) )
37		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` w ) = ( N + 2 ) -> ( ( # ` w ) - 1 ) = ( ( N + 2 ) - 1 ) )
38	37	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) -> ( ( # ` w ) - 1 ) = ( ( N + 2 ) - 1 ) )
39		nn0cn 11302	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. NN0 -> N e. CC )
40		2cnd 11093	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. NN0 -> 2 e. CC )
41		1cnd 10056	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. NN0 -> 1 e. CC )
42	39, 40, 41	addsubassd 10412	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> ( ( N + 2 ) - 1 ) = ( N + ( 2 - 1 ) ) )
43		2m1e1 11135	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 2 - 1 ) = 1
44	43	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. NN0 -> ( 2 - 1 ) = 1 )
45	44	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> ( N + ( 2 - 1 ) ) = ( N + 1 ) )
46	42, 45	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> ( ( N + 2 ) - 1 ) = ( N + 1 ) )
47	38, 46	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . . 20 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> ( ( # ` w ) - 1 ) = ( N + 1 ) )
48	47	opeq2d 4409	. . . . . . . . . . . . . . . . . . 19 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> <. 0 , ( ( # ` w ) - 1 ) >. = <. 0 , ( N + 1 ) >. )
49	48	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) = ( w substr <. 0 , ( N + 1 ) >. ) )
50	49	oveq1d 6665	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) = ( ( w substr <. 0 , ( N + 1 ) >. ) ++ <" ( lastS ` w ) "> ) )
51		swrdccatwrd 13468	. . . . . . . . . . . . . . . . . 18 |- ( ( w e. Word V /\ w =/= (/) ) -> ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) = w )
52	7, 20, 51	syl2an2 875	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> ( ( w substr <. 0 , ( ( # ` w ) - 1 ) >. ) ++ <" ( lastS ` w ) "> ) = w )
53	50, 52	eqtr3d 2658	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) -> ( ( w substr <. 0 , ( N + 1 ) >. ) ++ <" ( lastS ` w ) "> ) = w )
54	53	adantrr 753	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) ) -> ( ( w substr <. 0 , ( N + 1 ) >. ) ++ <" ( lastS ` w ) "> ) = w )
55	36, 54	eqtr2d 2657	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) ) -> w = ( W ++ <" ( lastS ` w ) "> ) )
56		simprrr 805	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) ) -> { ( lastS ` W ) , ( lastS ` w ) } e. E )
57		wwlksnextbij0.e	. . . . . . . . . . . . . . 15 |- E = (Edg ` G )
58	4, 57	wwlksnextbi 26789	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN0 /\ ( lastS ` w ) e. V ) /\ ( W e. Word V /\ w = ( W ++ <" ( lastS ` w ) "> ) /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) -> ( w e. ( ( N + 1 ) WWalksN G ) <-> W e. ( N WWalksN G ) ) )
59	6, 23, 32, 55, 56, 58	syl23anc 1333	. . . . . . . . . . . . 13 |- ( ( N e. NN0 /\ ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) ) -> ( w e. ( ( N + 1 ) WWalksN G ) <-> W e. ( N WWalksN G ) ) )
60	59	exbiri 652	. . . . . . . . . . . 12 |- ( N e. NN0 -> ( ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) -> ( W e. ( N WWalksN G ) -> w e. ( ( N + 1 ) WWalksN G ) ) ) )
61	60	com23 86	. . . . . . . . . . 11 |- ( N e. NN0 -> ( W e. ( N WWalksN G ) -> ( ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) -> w e. ( ( N + 1 ) WWalksN G ) ) ) )
62	61	3ad2ant2 1083	. . . . . . . . . 10 |- ( ( G e. _V /\ N e. NN0 /\ W e. Word V ) -> ( W e. ( N WWalksN G ) -> ( ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) -> w e. ( ( N + 1 ) WWalksN G ) ) ) )
63	5, 62	mpcom 38	. . . . . . . . 9 |- ( W e. ( N WWalksN G ) -> ( ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) -> w e. ( ( N + 1 ) WWalksN G ) ) )
64	63	expcomd 454	. . . . . . . 8 |- ( W e. ( N WWalksN G ) -> ( ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) -> ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) -> w e. ( ( N + 1 ) WWalksN G ) ) ) )
65	64	imp 445	. . . . . . 7 |- ( ( W e. ( N WWalksN G ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) -> ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) -> w e. ( ( N + 1 ) WWalksN G ) ) )
66	4, 57	wwlknp 26734	. . . . . . . . . . . 12 |- ( w e. ( ( N + 1 ) WWalksN G ) -> ( w e. Word V /\ ( # ` w ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E ) )
67	39, 41, 41	addassd 10062	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> ( ( N + 1 ) + 1 ) = ( N + ( 1 + 1 ) ) )
68		1p1e2 11134	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 1 + 1 ) = 2
69	68	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> ( 1 + 1 ) = 2 )
70	69	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> ( N + ( 1 + 1 ) ) = ( N + 2 ) )
71	67, 70	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN0 -> ( ( N + 1 ) + 1 ) = ( N + 2 ) )
72	71	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 |- ( N e. NN0 -> ( ( # ` w ) = ( ( N + 1 ) + 1 ) <-> ( # ` w ) = ( N + 2 ) ) )
73	72	biimpd 219	. . . . . . . . . . . . . . . . 17 |- ( N e. NN0 -> ( ( # ` w ) = ( ( N + 1 ) + 1 ) -> ( # ` w ) = ( N + 2 ) ) )
74	73	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ W e. Word V ) -> ( ( # ` w ) = ( ( N + 1 ) + 1 ) -> ( # ` w ) = ( N + 2 ) ) )
75	74	com12 32	. . . . . . . . . . . . . . 15 |- ( ( # ` w ) = ( ( N + 1 ) + 1 ) -> ( ( N e. NN0 /\ W e. Word V ) -> ( # ` w ) = ( N + 2 ) ) )
76	75	adantl 482	. . . . . . . . . . . . . 14 |- ( ( w e. Word V /\ ( # ` w ) = ( ( N + 1 ) + 1 ) ) -> ( ( N e. NN0 /\ W e. Word V ) -> ( # ` w ) = ( N + 2 ) ) )
77		simpl 473	. . . . . . . . . . . . . 14 |- ( ( w e. Word V /\ ( # ` w ) = ( ( N + 1 ) + 1 ) ) -> w e. Word V )
78	76, 77	jctild 566	. . . . . . . . . . . . 13 |- ( ( w e. Word V /\ ( # ` w ) = ( ( N + 1 ) + 1 ) ) -> ( ( N e. NN0 /\ W e. Word V ) -> ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) )
79	78	3adant3 1081	. . . . . . . . . . . 12 |- ( ( w e. Word V /\ ( # ` w ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E ) -> ( ( N e. NN0 /\ W e. Word V ) -> ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) )
80	66, 79	syl 17	. . . . . . . . . . 11 |- ( w e. ( ( N + 1 ) WWalksN G ) -> ( ( N e. NN0 /\ W e. Word V ) -> ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) )
81	80	com12 32	. . . . . . . . . 10 |- ( ( N e. NN0 /\ W e. Word V ) -> ( w e. ( ( N + 1 ) WWalksN G ) -> ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) )
82	81	3adant1 1079	. . . . . . . . 9 |- ( ( G e. _V /\ N e. NN0 /\ W e. Word V ) -> ( w e. ( ( N + 1 ) WWalksN G ) -> ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) )
83	5, 82	syl 17	. . . . . . . 8 |- ( W e. ( N WWalksN G ) -> ( w e. ( ( N + 1 ) WWalksN G ) -> ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) )
84	83	adantr 481	. . . . . . 7 |- ( ( W e. ( N WWalksN G ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) -> ( w e. ( ( N + 1 ) WWalksN G ) -> ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) ) )
85	65, 84	impbid 202	. . . . . 6 |- ( ( W e. ( N WWalksN G ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) -> ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) <-> w e. ( ( N + 1 ) WWalksN G ) ) )
86	85	ex 450	. . . . 5 |- ( W e. ( N WWalksN G ) -> ( ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) -> ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) <-> w e. ( ( N + 1 ) WWalksN G ) ) ) )
87	86	pm5.32rd 672	. . . 4 |- ( W e. ( N WWalksN G ) -> ( ( ( w e. Word V /\ ( # ` w ) = ( N + 2 ) ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) <-> ( w e. ( ( N + 1 ) WWalksN G ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) ) )
88	3, 87	syl5bb 272	. . 3 |- ( W e. ( N WWalksN G ) -> ( ( w e. Word V /\ ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) <-> ( w e. ( ( N + 1 ) WWalksN G ) /\ ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) ) ) )
89	88	rabbidva2 3186	. 2 |- ( W e. ( N WWalksN G ) -> { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } = { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } )
90	1, 89	syl5eq 2668	1 |- ( W e. ( N WWalksN G ) -> D = { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   (/)c0 3915   {cpr 4179   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   2c2 11070   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939   WWalksN cwwlksn 26718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wwlksnextsur  26795  wwlksnextbij  26797