Theorem wwlksnredwwlkn0 26791

Description: For each walk (as word) of length at least 1 there is a shorter walk (as word) starting at the same vertex. (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
wwlksnredwwlkn0	|- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( ( W ` 0 ) = P <-> E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) )

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

Proof of Theorem wwlksnredwwlkn0

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnredwwlkn.e	. . . . 5 |- E = (Edg ` G )
2	1	wwlksnredwwlkn 26790	. . . 4 |- ( 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 ) ) )
3	2	imp 445	. . 3 |- ( ( 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 ) )
4		simpl 473	. . . . . . . . 9 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) -> ( W substr <. 0 , ( N + 1 ) >. ) = y )
5	4	adantl 482	. . . . . . . 8 |- ( ( ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) /\ ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) -> ( W substr <. 0 , ( N + 1 ) >. ) = y )
6		fveq1 6190	. . . . . . . . . . . . . 14 |- ( y = ( W substr <. 0 , ( N + 1 ) >. ) -> ( y ` 0 ) = ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) )
7	6	eqcoms 2630	. . . . . . . . . . . . 13 |- ( ( W substr <. 0 , ( N + 1 ) >. ) = y -> ( y ` 0 ) = ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) )
8	7	adantr 481	. . . . . . . . . . . 12 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) ) -> ( y ` 0 ) = ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) )
9		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- (Vtx ` G ) = (Vtx ` G )
10	9, 1	wwlknp 26734	. . . . . . . . . . . . . . . . . 18 |- ( 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 ) )
11		nn0p1nn 11332	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. NN0 -> ( N + 1 ) e. NN )
12		peano2nn0 11333	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
13		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N + 1 ) e. NN0 -> ( N + 1 ) e. RR )
14		lep1 10862	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N + 1 ) e. RR -> ( N + 1 ) <_ ( ( N + 1 ) + 1 ) )
15	12, 13, 14	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. NN0 -> ( N + 1 ) <_ ( ( N + 1 ) + 1 ) )
16		peano2nn0 11333	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N + 1 ) e. NN0 -> ( ( N + 1 ) + 1 ) e. NN0 )
17	16	nn0zd 11480	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N + 1 ) e. NN0 -> ( ( N + 1 ) + 1 ) e. ZZ )
18		fznn 12408	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( N + 1 ) + 1 ) e. ZZ -> ( ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) <-> ( ( N + 1 ) e. NN /\ ( N + 1 ) <_ ( ( N + 1 ) + 1 ) ) ) )
19	12, 17, 18	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. NN0 -> ( ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) <-> ( ( N + 1 ) e. NN /\ ( N + 1 ) <_ ( ( N + 1 ) + 1 ) ) ) )
20	11, 15, 19	mpbir2and 957	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) )
21		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( 1 ... ( # ` W ) ) = ( 1 ... ( ( N + 1 ) + 1 ) ) )
22	21	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( ( N + 1 ) e. ( 1 ... ( # ` W ) ) <-> ( N + 1 ) e. ( 1 ... ( ( N + 1 ) + 1 ) ) ) )
23	20, 22	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( N e. NN0 -> ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) )
24	23	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( N e. NN0 -> ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) )
25		simpl 473	. . . . . . . . . . . . . . . . . . . 20 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> W e. Word (Vtx ` G ) )
26	24, 25	jctild 566	. . . . . . . . . . . . . . . . . . 19 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( N e. NN0 -> ( W e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) ) )
27	26	3adant3 1081	. . . . . . . . . . . . . . . . . 18 |- ( ( 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 e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) ) )
28	10, 27	syl 17	. . . . . . . . . . . . . . . . 17 |- ( W e. ( ( N + 1 ) WWalksN G ) -> ( N e. NN0 -> ( W e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) ) )
29	28	impcom 446	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( W e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) )
30	29	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) -> ( W e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) )
31	30	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) -> ( W e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) )
32	31	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) ) -> ( W e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) )
33		swrd0fv0 13440	. . . . . . . . . . . . 13 |- ( ( W e. Word (Vtx ` G ) /\ ( N + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( W ` 0 ) )
34	32, 33	syl 17	. . . . . . . . . . . 12 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( W ` 0 ) )
35		simprll 802	. . . . . . . . . . . 12 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) ) -> ( W ` 0 ) = P )
36	8, 34, 35	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) ) -> ( y ` 0 ) = P )
37	36	ex 450	. . . . . . . . . 10 |- ( ( W substr <. 0 , ( N + 1 ) >. ) = y -> ( ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) -> ( y ` 0 ) = P ) )
38	37	adantr 481	. . . . . . . . 9 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) -> ( ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) -> ( y ` 0 ) = P ) )
39	38	impcom 446	. . . . . . . 8 |- ( ( ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) /\ ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) -> ( y ` 0 ) = P )
40		simpr 477	. . . . . . . . 9 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) -> { ( lastS ` y ) , ( lastS ` W ) } e. E )
41	40	adantl 482	. . . . . . . 8 |- ( ( ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) /\ ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) -> { ( lastS ` y ) , ( lastS ` W ) } e. E )
42	5, 39, 41	3jca 1242	. . . . . . 7 |- ( ( ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) /\ ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) )
43	42	ex 450	. . . . . 6 |- ( ( ( ( W ` 0 ) = P /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) /\ y e. ( N WWalksN G ) ) -> ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) )
44	43	reximdva 3017	. . . . 5 |- ( ( ( W ` 0 ) = P /\ ( 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 ) -> E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) )
45	44	ex 450	. . . 4 |- ( ( W ` 0 ) = P -> ( ( 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 ) -> E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) ) )
46	45	com13 88	. . 3 |- ( E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) -> ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( ( W ` 0 ) = P -> E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) ) )
47	3, 46	mpcom 38	. 2 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( ( W ` 0 ) = P -> E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) ) )
48	29, 33	syl 17	. . . . . . . . 9 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( W ` 0 ) )
49	48	eqcomd 2628	. . . . . . . 8 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( W ` 0 ) = ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) )
50	49	adantl 482	. . . . . . 7 |- ( ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P ) /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) -> ( W ` 0 ) = ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) )
51		fveq1 6190	. . . . . . . . 9 |- ( ( W substr <. 0 , ( N + 1 ) >. ) = y -> ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( y ` 0 ) )
52	51	adantr 481	. . . . . . . 8 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( y ` 0 ) )
53	52	adantr 481	. . . . . . 7 |- ( ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P ) /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) ` 0 ) = ( y ` 0 ) )
54		simpr 477	. . . . . . . 8 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P ) -> ( y ` 0 ) = P )
55	54	adantr 481	. . . . . . 7 |- ( ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P ) /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) -> ( y ` 0 ) = P )
56	50, 53, 55	3eqtrd 2660	. . . . . 6 |- ( ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P ) /\ ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) ) -> ( W ` 0 ) = P )
57	56	ex 450	. . . . 5 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P ) -> ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( W ` 0 ) = P ) )
58	57	3adant3 1081	. . . 4 |- ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) -> ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( W ` 0 ) = P ) )
59	58	com12 32	. . 3 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) -> ( W ` 0 ) = P ) )
60	59	rexlimdvw 3034	. 2 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` W ) } e. E ) -> ( W ` 0 ) = P ) )
61	47, 60	impbid 202	1 |- ( ( N e. NN0 /\ W e. ( ( N + 1 ) WWalksN G ) ) -> ( ( W ` 0 ) = P <-> E. y e. ( N WWalksN G ) ( ( W substr <. 0 , ( N + 1 ) >. ) = y /\ ( y ` 0 ) = P /\ { ( 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   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...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:  rusgrnumwwlks  26869