Theorem wwlksnredwwlkn 26790

Description: For each walk (as word) of length at least 1 there is a shorter walk (as word). (Contributed by Alexander van der Vekens, 22-Aug-2018.) (Revised by AV, 18-Apr-2021.)

Hypothesis

Ref	Expression
wwlksnredwwlkn.e	|- E = (Edg ` G )

Assertion

Ref	Expression
wwlksnredwwlkn	|- ( N e. NN0 -> ( W e. ( ( N + 1 ) WWalksN G ) -> E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) )

Distinct variable groups:   y, E   y, G   y, N   y, W

Proof of Theorem wwlksnredwwlkn

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2623	. . 3 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( W substr <. 0 , ( N + 1 ) >. ) = ( W substr <. 0 , ( N + 1 ) >. ) )
2		eqid 2622	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
3		wwlksnredwwlkn.e	. . . . 5 |- E = (Edg ` G )
4	2, 3	wwlknp 26734	. . . 4 |- ( W e. ( ( N + 1 ) WWalksN G ) -> ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) )
5		simprl 794	. . . . . . . . 9 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) ) -> W e. Word (Vtx ` G ) )
6		peano2nn0 11333	. . . . . . . . . . . . 13 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
7		peano2nn0 11333	. . . . . . . . . . . . 13 |- ( ( N + 1 ) e. NN0 -> ( ( N + 1 ) + 1 ) e. NN0 )
8	6, 7	syl 17	. . . . . . . . . . . 12 |- ( N e. NN0 -> ( ( N + 1 ) + 1 ) e. NN0 )
9		id 22	. . . . . . . . . . . . 13 |- ( N e. NN0 -> N e. NN0 )
10		nn0p1nn 11332	. . . . . . . . . . . . . 14 |- ( ( N + 1 ) e. NN0 -> ( ( N + 1 ) + 1 ) e. NN )
11	6, 10	syl 17	. . . . . . . . . . . . 13 |- ( N e. NN0 -> ( ( N + 1 ) + 1 ) e. NN )
12		nn0re 11301	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> N e. RR )
13		id 22	. . . . . . . . . . . . . . . 16 |- ( N e. RR -> N e. RR )
14		peano2re 10209	. . . . . . . . . . . . . . . 16 |- ( N e. RR -> ( N + 1 ) e. RR )
15		peano2re 10209	. . . . . . . . . . . . . . . . 17 |- ( ( N + 1 ) e. RR -> ( ( N + 1 ) + 1 ) e. RR )
16	14, 15	syl 17	. . . . . . . . . . . . . . . 16 |- ( N e. RR -> ( ( N + 1 ) + 1 ) e. RR )
17	13, 14, 16	3jca 1242	. . . . . . . . . . . . . . 15 |- ( N e. RR -> ( N e. RR /\ ( N + 1 ) e. RR /\ ( ( N + 1 ) + 1 ) e. RR ) )
18	12, 17	syl 17	. . . . . . . . . . . . . 14 |- ( N e. NN0 -> ( N e. RR /\ ( N + 1 ) e. RR /\ ( ( N + 1 ) + 1 ) e. RR ) )
19	12	ltp1d 10954	. . . . . . . . . . . . . 14 |- ( N e. NN0 -> N < ( N + 1 ) )
20		nn0re 11301	. . . . . . . . . . . . . . . 16 |- ( ( N + 1 ) e. NN0 -> ( N + 1 ) e. RR )
21	6, 20	syl 17	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> ( N + 1 ) e. RR )
22	21	ltp1d 10954	. . . . . . . . . . . . . 14 |- ( N e. NN0 -> ( N + 1 ) < ( ( N + 1 ) + 1 ) )
23		lttr 10114	. . . . . . . . . . . . . . 15 |- ( ( N e. RR /\ ( N + 1 ) e. RR /\ ( ( N + 1 ) + 1 ) e. RR ) -> ( ( N < ( N + 1 ) /\ ( N + 1 ) < ( ( N + 1 ) + 1 ) ) -> N < ( ( N + 1 ) + 1 ) ) )
24	23	imp 445	. . . . . . . . . . . . . 14 |- ( ( ( N e. RR /\ ( N + 1 ) e. RR /\ ( ( N + 1 ) + 1 ) e. RR ) /\ ( N < ( N + 1 ) /\ ( N + 1 ) < ( ( N + 1 ) + 1 ) ) ) -> N < ( ( N + 1 ) + 1 ) )
25	18, 19, 22, 24	syl12anc 1324	. . . . . . . . . . . . 13 |- ( N e. NN0 -> N < ( ( N + 1 ) + 1 ) )
26		elfzo0 12508	. . . . . . . . . . . . 13 |- ( N e. ( 0..^ ( ( N + 1 ) + 1 ) ) <-> ( N e. NN0 /\ ( ( N + 1 ) + 1 ) e. NN /\ N < ( ( N + 1 ) + 1 ) ) )
27	9, 11, 25, 26	syl3anbrc 1246	. . . . . . . . . . . 12 |- ( N e. NN0 -> N e. ( 0..^ ( ( N + 1 ) + 1 ) ) )
28		fz0add1fz1 12537	. . . . . . . . . . . 12 |- ( ( ( ( N + 1 ) + 1 ) e. NN0 /\ N e. ( 0..^ ( ( N + 1 ) + 1 ) ) ) -> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) )
29	8, 27, 28	syl2anc 693	. . . . . . . . . . 11 |- ( N e. NN0 -> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) )
30	29	adantr 481	. . . . . . . . . 10 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) ) -> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) )
31		oveq2 6658	. . . . . . . . . . . . 13 |- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( 1 ... ( # ` W ) ) = ( 1 ... ( ( N + 1 ) + 1 ) ) )
32	31	eleq2d 2687	. . . . . . . . . . . 12 |- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( ( N + 1 ) e. ( 1 ... ( # ` W ) ) <-> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) ) )
33	32	adantl 482	. . . . . . . . . . 11 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( ( N + 1 ) e. ( 1 ... ( # ` W ) ) <-> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) ) )
34	33	adantl 482	. . . . . . . . . 10 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) ) -> ( ( N + 1 ) e. ( 1 ... ( # ` W ) ) <-> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) ) )
35	30, 34	mpbird 247	. . . . . . . . 9 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) ) -> ( N + 1 ) e. ( 1 ... ( # ` W ) ) )
36	5, 35	jca 554	. . . . . . . 8 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) ) -> ( W e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) )
37	36	3adantr3 1222	. . . . . . 7 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> ( W e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) )
38		swrd0fvlsw 13443	. . . . . . 7 |- ( ( W e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) = ( W ` ( ( N + 1 ) - 1 ) ) )
39	37, 38	syl 17	. . . . . 6 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) = ( W ` ( ( N + 1 ) - 1 ) ) )
40		lsw 13351	. . . . . . . 8 |- ( W e. Word (Vtx ` G ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
41	40	3ad2ant1 1082	. . . . . . 7 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
42	41	adantl 482	. . . . . 6 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
43	39, 42	preq12d 4276	. . . . 5 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` W ) } = { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( # ` W ) - 1 ) ) } )
44		oveq1 6657	. . . . . . . . . . 11 |- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( ( # ` W ) - 1 ) = ( ( ( N + 1 ) + 1 ) - 1 ) )
45	44	3ad2ant2 1083	. . . . . . . . . 10 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> ( ( # ` W ) - 1 ) = ( ( ( N + 1 ) + 1 ) - 1 ) )
46	45	adantl 482	. . . . . . . . 9 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> ( ( # ` W ) - 1 ) = ( ( ( N + 1 ) + 1 ) - 1 ) )
47	46	fveq2d 6195	. . . . . . . 8 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) )
48	47	preq2d 4275	. . . . . . 7 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( # ` W ) - 1 ) ) } = { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) } )
49		nn0cn 11302	. . . . . . . . . . 11 |- ( N e. NN0 -> N e. CC )
50		1cnd 10056	. . . . . . . . . . 11 |- ( N e. NN0 -> 1 e. CC )
51	49, 50	pncand 10393	. . . . . . . . . 10 |- ( N e. NN0 -> ( ( N + 1 ) - 1 ) = N )
52	51	fveq2d 6195	. . . . . . . . 9 |- ( N e. NN0 -> ( W ` ( ( N + 1 ) - 1 ) ) = ( W ` N ) )
53	6	nn0cnd 11353	. . . . . . . . . . 11 |- ( N e. NN0 -> ( N + 1 ) e. CC )
54	53, 50	pncand 10393	. . . . . . . . . 10 |- ( N e. NN0 -> ( ( ( N + 1 ) + 1 ) - 1 ) = ( N + 1 ) )
55	54	fveq2d 6195	. . . . . . . . 9 |- ( N e. NN0 -> ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) = ( W ` ( N + 1 ) ) )
56	52, 55	preq12d 4276	. . . . . . . 8 |- ( N e. NN0 -> { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) } = { ( W ` N ) , ( W ` ( N + 1 ) ) } )
57	56	adantr 481	. . . . . . 7 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( ( N + 1 ) + 1 ) - 1 ) ) } = { ( W ` N ) , ( W ` ( N + 1 ) ) } )
58	48, 57	eqtrd 2656	. . . . . 6 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( # ` W ) - 1 ) ) } = { ( W ` N ) , ( W ` ( N + 1 ) ) } )
59		fveq2 6191	. . . . . . . . . . . 12 |- ( i = N -> ( W ` i ) = ( W ` N ) )
60		oveq1 6657	. . . . . . . . . . . . 13 |- ( i = N -> ( i + 1 ) = ( N + 1 ) )
61	60	fveq2d 6195	. . . . . . . . . . . 12 |- ( i = N -> ( W ` ( i + 1 ) ) = ( W ` ( N + 1 ) ) )
62	59, 61	preq12d 4276	. . . . . . . . . . 11 |- ( i = N -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` N ) , ( W ` ( N + 1 ) ) } )
63	62	eleq1d 2686	. . . . . . . . . 10 |- ( i = N -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E <-> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) )
64	63	rspcv 3305	. . . . . . . . 9 |- ( N e. ( 0..^ ( N + 1 ) ) -> ( A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E -> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) )
65		fzonn0p1 12544	. . . . . . . . 9 |- ( N e. NN0 -> N e. ( 0..^ ( N + 1 ) ) )
66	64, 65	syl11 33	. . . . . . . 8 |- ( A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E -> ( N e. NN0 -> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) )
67	66	3ad2ant3 1084	. . . . . . 7 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> ( N e. NN0 -> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E ) )
68	67	impcom 446	. . . . . 6 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( W ` N ) , ( W ` ( N + 1 ) ) } e. E )
69	58, 68	eqeltrd 2701	. . . . 5 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( W ` ( ( N + 1 ) - 1 ) ) , ( W ` ( ( # ` W ) - 1 ) ) } e. E )
70	43, 69	eqeltrd 2701	. . . 4 |- ( ( N e. NN0 /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) ) -> { ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` W ) } e. E )
71	4, 70	sylan2 491	. . 3 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> { ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` W ) } e. E )
72		wwlksnred 26787	. . . . 5 |- ( N e. NN0 -> ( W e. ( ( N + 1 ) WWalksN G ) -> ( W substr <. 0 , ( N + 1 ) >. ) e. ( N WWalksN G ) ) )
73	72	imp 445	. . . 4 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( W substr <. 0 , ( N + 1 ) >. ) e. ( N WWalksN G ) )
74		eqeq2 2633	. . . . . 6 |- ( y = ( W substr <. 0 , ( N + 1 ) >. ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) = y <-> ( W substr <. 0 , ( N + 1 ) >. ) = ( W substr <. 0 , ( N + 1 ) >. ) ) )
75		fveq2 6191	. . . . . . . 8 |- ( y = ( W substr <. 0 , ( N + 1 ) >. ) -> ( lastS ` y ) = ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) )
76	75	preq1d 4274	. . . . . . 7 |- ( y = ( W substr <. 0 , ( N + 1 ) >. ) -> { ( lastS ` y ) , ( lastS ` W ) } = { ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` W ) } )
77	76	eleq1d 2686	. . . . . 6 |- ( y = ( W substr <. 0 , ( N + 1 ) >. ) -> ( { ( lastS ` y ) , ( lastS ` W ) } e. E <-> { ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` W ) } e. E ) )
78	74, 77	anbi12d 747	. . . . 5 |- ( y = ( W substr <. 0 , ( N + 1 ) >. ) -> ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) <-> ( ( W substr <. 0 , ( N + 1 ) >. ) = ( W substr <. 0 , ( N + 1 ) >. ) /\ { ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` W ) } e. E ) ) )
79	78	adantl 482	. . . 4 |- ( ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) /\ y = ( W substr <. 0 , ( N + 1 ) >. ) ) -> ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) <-> ( ( W substr <. 0 , ( N + 1 ) >. ) = ( W substr <. 0 , ( N + 1 ) >. ) /\ { ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` W ) } e. E ) ) )
80	73, 79	rspcedv 3313	. . 3 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( ( ( W substr <. 0 , ( N + 1 ) >. ) = ( W substr <. 0 , ( N + 1 ) >. ) /\ { ( lastS ` ( W substr <. 0 , ( N + 1 ) >. ) ) , ( lastS ` W ) } e. E ) -> E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) )
81	1, 71, 80	mp2and 715	. 2 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) )
82	81	ex 450	1 |- ( N e. NN0 -> ( W e. ( ( N + 1 ) WWalksN G ) -> E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {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   NNcn 11020   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-substr 13303  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wwlksnredwwlkn0  26791