Theorem rusgrnumwwlkl1 26863

Description: In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.)

Hypothesis

Ref	Expression
rusgrnumwwlkl1.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
rusgrnumwwlkl1	|- ( ( G RegUSGraph K /\ P e. V ) -> ( # ` { w e. ( 1 WWalksN G ) | ( w ` 0 ) = P } ) = K )

Distinct variable groups:   w, G   w, K   w, P   w, V

Proof of Theorem rusgrnumwwlkl1

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nn0 11308	. . . . . . . . 9 |- 1 e. NN0
2		iswwlksn 26730	. . . . . . . . 9 |- ( 1 e. NN0 -> ( w e. ( 1 WWalksN G ) <-> ( w e. (WWalks ` G ) /\ ( # ` w ) = ( 1 + 1 ) ) ) )
3	1, 2	ax-mp 5	. . . . . . . 8 |- ( w e. ( 1 WWalksN G ) <-> ( w e. (WWalks ` G ) /\ ( # ` w ) = ( 1 + 1 ) ) )
4		rusgrnumwwlkl1.v	. . . . . . . . . 10 |- V = (Vtx ` G )
5		eqid 2622	. . . . . . . . . 10 |- (Edg ` G ) = (Edg ` G )
6	4, 5	iswwlks 26728	. . . . . . . . 9 |- ( w e. (WWalks ` G ) <-> ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) )
7	6	anbi1i 731	. . . . . . . 8 |- ( ( w e. (WWalks ` G ) /\ ( # ` w ) = ( 1 + 1 ) ) <-> ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( 1 + 1 ) ) )
8	3, 7	bitri 264	. . . . . . 7 |- ( w e. ( 1 WWalksN G ) <-> ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( 1 + 1 ) ) )
9	8	a1i 11	. . . . . 6 |- ( ( G RegUSGraph K /\ P e. V ) -> ( w e. ( 1 WWalksN G ) <-> ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( 1 + 1 ) ) ) )
10	9	anbi1d 741	. . . . 5 |- ( ( G RegUSGraph K /\ P e. V ) -> ( ( w e. ( 1 WWalksN G ) /\ ( w ` 0 ) = P ) <-> ( ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( 1 + 1 ) ) /\ ( w ` 0 ) = P ) ) )
11		1p1e2 11134	. . . . . . . . . . 11 |- ( 1 + 1 ) = 2
12	11	eqeq2i 2634	. . . . . . . . . 10 |- ( ( # ` w ) = ( 1 + 1 ) <-> ( # ` w ) = 2 )
13	12	a1i 11	. . . . . . . . 9 |- ( ( G RegUSGraph K /\ P e. V ) -> ( ( # ` w ) = ( 1 + 1 ) <-> ( # ` w ) = 2 ) )
14	13	anbi2d 740	. . . . . . . 8 |- ( ( G RegUSGraph K /\ P e. V ) -> ( ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( 1 + 1 ) ) <-> ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = 2 ) ) )
15		3anass 1042	. . . . . . . . . . . 12 |- ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) <-> ( w =/= (/) /\ ( w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
16	15	a1i 11	. . . . . . . . . . 11 |- ( ( ( G RegUSGraph K /\ P e. V ) /\ ( # ` w ) = 2 ) -> ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) <-> ( w =/= (/) /\ ( w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) ) ) )
17		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( w = (/) -> ( # ` w ) = ( # ` (/) ) )
18		hash0 13158	. . . . . . . . . . . . . . . 16 |- ( # ` (/) ) = 0
19	17, 18	syl6eq 2672	. . . . . . . . . . . . . . 15 |- ( w = (/) -> ( # ` w ) = 0 )
20		2ne0 11113	. . . . . . . . . . . . . . . . 17 |- 2 =/= 0
21	20	nesymi 2851	. . . . . . . . . . . . . . . 16 |- -. 0 = 2
22		eqeq1 2626	. . . . . . . . . . . . . . . 16 |- ( ( # ` w ) = 0 -> ( ( # ` w ) = 2 <-> 0 = 2 ) )
23	21, 22	mtbiri 317	. . . . . . . . . . . . . . 15 |- ( ( # ` w ) = 0 -> -. ( # ` w ) = 2 )
24	19, 23	syl 17	. . . . . . . . . . . . . 14 |- ( w = (/) -> -. ( # ` w ) = 2 )
25	24	necon2ai 2823	. . . . . . . . . . . . 13 |- ( ( # ` w ) = 2 -> w =/= (/) )
26	25	adantl 482	. . . . . . . . . . . 12 |- ( ( ( G RegUSGraph K /\ P e. V ) /\ ( # ` w ) = 2 ) -> w =/= (/) )
27	26	biantrurd 529	. . . . . . . . . . 11 |- ( ( ( G RegUSGraph K /\ P e. V ) /\ ( # ` w ) = 2 ) -> ( ( w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) <-> ( w =/= (/) /\ ( w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) ) ) )
28		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( ( # ` w ) = 2 -> ( ( # ` w ) - 1 ) = ( 2 - 1 ) )
29		2m1e1 11135	. . . . . . . . . . . . . . . . 17 |- ( 2 - 1 ) = 1
30	28, 29	syl6eq 2672	. . . . . . . . . . . . . . . 16 |- ( ( # ` w ) = 2 -> ( ( # ` w ) - 1 ) = 1 )
31	30	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( ( # ` w ) = 2 -> ( 0..^ ( ( # ` w ) - 1 ) ) = ( 0..^ 1 ) )
32	31	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( G RegUSGraph K /\ P e. V ) /\ ( # ` w ) = 2 ) -> ( 0..^ ( ( # ` w ) - 1 ) ) = ( 0..^ 1 ) )
33	32	raleqdv 3144	. . . . . . . . . . . . 13 |- ( ( ( G RegUSGraph K /\ P e. V ) /\ ( # ` w ) = 2 ) -> ( A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. ( 0..^ 1 ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) )
34		fzo01 12550	. . . . . . . . . . . . . . 15 |- ( 0..^ 1 ) = { 0 }
35	34	raleqi 3142	. . . . . . . . . . . . . 14 |- ( A. i e. ( 0..^ 1 ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. { 0 } { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) )
36		c0ex 10034	. . . . . . . . . . . . . . 15 |- 0 e. _V
37		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( i = 0 -> ( w ` i ) = ( w ` 0 ) )
38		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
39		0p1e1 11132	. . . . . . . . . . . . . . . . . . 19 |- ( 0 + 1 ) = 1
40	38, 39	syl6eq 2672	. . . . . . . . . . . . . . . . . 18 |- ( i = 0 -> ( i + 1 ) = 1 )
41	40	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( i = 0 -> ( w ` ( i + 1 ) ) = ( w ` 1 ) )
42	37, 41	preq12d 4276	. . . . . . . . . . . . . . . 16 |- ( i = 0 -> { ( w ` i ) , ( w ` ( i + 1 ) ) } = { ( w ` 0 ) , ( w ` 1 ) } )
43	42	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( i = 0 -> ( { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
44	36, 43	ralsn 4222	. . . . . . . . . . . . . 14 |- ( A. i e. { 0 } { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) )
45	35, 44	bitri 264	. . . . . . . . . . . . 13 |- ( A. i e. ( 0..^ 1 ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) )
46	33, 45	syl6bb 276	. . . . . . . . . . . 12 |- ( ( ( G RegUSGraph K /\ P e. V ) /\ ( # ` w ) = 2 ) -> ( A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
47	46	anbi2d 740	. . . . . . . . . . 11 |- ( ( ( G RegUSGraph K /\ P e. V ) /\ ( # ` w ) = 2 ) -> ( ( w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) <-> ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) ) )
48	16, 27, 47	3bitr2d 296	. . . . . . . . . 10 |- ( ( ( G RegUSGraph K /\ P e. V ) /\ ( # ` w ) = 2 ) -> ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) <-> ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) ) )
49	48	ex 450	. . . . . . . . 9 |- ( ( G RegUSGraph K /\ P e. V ) -> ( ( # ` w ) = 2 -> ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) <-> ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) ) ) )
50	49	pm5.32rd 672	. . . . . . . 8 |- ( ( G RegUSGraph K /\ P e. V ) -> ( ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = 2 ) <-> ( ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( # ` w ) = 2 ) ) )
51	14, 50	bitrd 268	. . . . . . 7 |- ( ( G RegUSGraph K /\ P e. V ) -> ( ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( 1 + 1 ) ) <-> ( ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( # ` w ) = 2 ) ) )
52	51	anbi1d 741	. . . . . 6 |- ( ( G RegUSGraph K /\ P e. V ) -> ( ( ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( 1 + 1 ) ) /\ ( w ` 0 ) = P ) <-> ( ( ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( # ` w ) = 2 ) /\ ( w ` 0 ) = P ) ) )
53		anass 681	. . . . . 6 |- ( ( ( ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( # ` w ) = 2 ) /\ ( w ` 0 ) = P ) <-> ( ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) ) )
54	52, 53	syl6bb 276	. . . . 5 |- ( ( G RegUSGraph K /\ P e. V ) -> ( ( ( ( w =/= (/) /\ w e. Word V /\ A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` w ) = ( 1 + 1 ) ) /\ ( w ` 0 ) = P ) <-> ( ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) ) ) )
55		anass 681	. . . . . . 7 |- ( ( ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) ) <-> ( w e. Word V /\ ( { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) ) ) )
56		ancom 466	. . . . . . . . 9 |- ( ( { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) ) <-> ( ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
57		df-3an 1039	. . . . . . . . 9 |- ( ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) <-> ( ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
58	56, 57	bitr4i 267	. . . . . . . 8 |- ( ( { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) ) <-> ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
59	58	anbi2i 730	. . . . . . 7 |- ( ( w e. Word V /\ ( { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) ) ) <-> ( w e. Word V /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) ) )
60	55, 59	bitri 264	. . . . . 6 |- ( ( ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) ) <-> ( w e. Word V /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) ) )
61	60	a1i 11	. . . . 5 |- ( ( G RegUSGraph K /\ P e. V ) -> ( ( ( w e. Word V /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P ) ) <-> ( w e. Word V /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) ) ) )
62	10, 54, 61	3bitrd 294	. . . 4 |- ( ( G RegUSGraph K /\ P e. V ) -> ( ( w e. ( 1 WWalksN G ) /\ ( w ` 0 ) = P ) <-> ( w e. Word V /\ ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) ) ) )
63	62	rabbidva2 3186	. . 3 |- ( ( G RegUSGraph K /\ P e. V ) -> { w e. ( 1 WWalksN G ) | ( w ` 0 ) = P } = { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) } )
64	63	fveq2d 6195	. 2 |- ( ( G RegUSGraph K /\ P e. V ) -> ( # ` { w e. ( 1 WWalksN G ) | ( w ` 0 ) = P } ) = ( # ` { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) } ) )
65	4	rusgrnumwrdl2 26482	. 2 |- ( ( G RegUSGraph K /\ P e. V ) -> ( # ` { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) } ) = K )
66	64, 65	eqtrd 2656	1 |- ( ( G RegUSGraph K /\ P e. V ) -> ( # ` { w e. ( 1 WWalksN G ) | ( w ` 0 ) = P } ) = K )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   (/)c0 3915   {csn 4177   {cpr 4179   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   2c2 11070   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  Edgcedg 25939   RegUSGraph crusgr 26452  WWalkscwwlks 26717   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-fal 1489  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-rmo 2920  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-2o 7561  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-cda 8990  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-xadd 11947  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-nbgr 26228  df-vtxdg 26362  df-rgr 26453  df-rusgr 26454  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  rusgrnumwwlkb1  26867