clwwlkinwwlk - Metamath Proof Explorer

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Theorem clwwlkinwwlk 26905

Description: If the initial vertex of a walk occurs another time in the walk, the walk starts with a closed walk. Since the walk is expressed as a word over vertices, the closed walk can be expressed as a subword of this word. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 23-Jan-2022.)

**Assertion**
Ref	Expression
clwwlkinwwlk	$/\$ $/\$ WWalksN $/\$ substr ClWWalksN

Proof of Theorem clwwlkinwwlk Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 Vtx Vtx
2		eqid 2622	. . 3 Edg Edg
3	1, 2	wwlknp 26734	. 2 WWalksN Word Vtx $/\$ $/\$ ..^ Edg
4		swrdcl 13419	. . . . . . . . . . . 12 Word Vtx substr Word Vtx
5	4	adantr 481	. . . . . . . . . . 11 Word Vtx $/\$ substr Word Vtx
6	5	adantr 481	. . . . . . . . . 10 Word Vtx $/\$ $/\$ $/\$ substr Word Vtx
7		simpll 790	. . . . . . . . . . 11 Word Vtx $/\$ $/\$ $/\$ Word Vtx
8		simprl 794	. . . . . . . . . . 11 Word Vtx $/\$ $/\$ $/\$
9		eluz2 11693	. . . . . . . . . . . . . . . 16 $/\$ $/\$
10		zre 11381	. . . . . . . . . . . . . . . . 17
11		zre 11381	. . . . . . . . . . . . . . . . 17
12		id 22	. . . . . . . . . . . . . . . . 17
13	10, 11, 12	3anim123i 1247	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
14	9, 13	sylbi 207	. . . . . . . . . . . . . . 15 $/\$ $/\$
15		letrp1 10865	. . . . . . . . . . . . . . 15 $/\$ $/\$
16	14, 15	syl 17	. . . . . . . . . . . . . 14
17	16	adantl 482	. . . . . . . . . . . . 13 $/\$
18	17	adantl 482	. . . . . . . . . . . 12 Word Vtx $/\$ $/\$ $/\$
19		breq2 4657	. . . . . . . . . . . . 13
20	19	ad2antlr 763	. . . . . . . . . . . 12 Word Vtx $/\$ $/\$ $/\$
21	18, 20	mpbird 247	. . . . . . . . . . 11 Word Vtx $/\$ $/\$ $/\$
22		swrdn0 13430	. . . . . . . . . . 11 Word Vtx $/\$ $/\$ substr
23	7, 8, 21, 22	syl3anc 1326	. . . . . . . . . 10 Word Vtx $/\$ $/\$ $/\$ substr
24	6, 23	jca 554	. . . . . . . . 9 Word Vtx $/\$ $/\$ $/\$ substr Word Vtx $/\$ substr
25	24	3adantl3 1219	. . . . . . . 8 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ substr Word Vtx $/\$ substr
26	25	adantr 481	. . . . . . 7 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ $/\$ substr Word Vtx $/\$ substr
27		nnz 11399	. . . . . . . . . . . . . . . . . 18
28		1nn0 11308	. . . . . . . . . . . . . . . . . 18
29		eluzmn 11694	. . . . . . . . . . . . . . . . . 18 $/\$
30	27, 28, 29	sylancl 694	. . . . . . . . . . . . . . . . 17
31		uzss 11708	. . . . . . . . . . . . . . . . 17
32	30, 31	syl 17	. . . . . . . . . . . . . . . 16
33	32	sselda 3603	. . . . . . . . . . . . . . 15 $/\$
34		fzoss2 12496	. . . . . . . . . . . . . . 15 ..^ ..^
35	33, 34	syl 17	. . . . . . . . . . . . . 14 $/\$ ..^ ..^
36	35	3ad2ant3 1084	. . . . . . . . . . . . 13 Word Vtx $/\$ $/\$ $/\$ ..^ ..^
37		ssralv 3666	. . . . . . . . . . . . 13 ..^ ..^ ..^ Edg ..^ Edg
38	36, 37	syl 17	. . . . . . . . . . . 12 Word Vtx $/\$ $/\$ $/\$ ..^ Edg ..^ Edg
39	38	3exp 1264	. . . . . . . . . . 11 Word Vtx $/\$ ..^ Edg ..^ Edg
40	39	com34 91	. . . . . . . . . 10 Word Vtx ..^ Edg $/\$ ..^ Edg
41	40	3imp1 1280	. . . . . . . . 9 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ ..^ Edg
42	41	adantr 481	. . . . . . . 8 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ $/\$ ..^ Edg
43		nnnn0 11299	. . . . . . . . . . . . . . . . . . . 20
44		elnn0uz 11725	. . . . . . . . . . . . . . . . . . . 20
45	43, 44	sylib 208	. . . . . . . . . . . . . . . . . . 19
46		eluzfz 12337	. . . . . . . . . . . . . . . . . . 19 $/\$
47	45, 46	sylan 488	. . . . . . . . . . . . . . . . . 18 $/\$
48		fzelp1 12393	. . . . . . . . . . . . . . . . . 18
49	47, 48	syl 17	. . . . . . . . . . . . . . . . 17 $/\$
50	49	adantl 482	. . . . . . . . . . . . . . . 16 Word Vtx $/\$ $/\$ $/\$
51		oveq2 6658	. . . . . . . . . . . . . . . . . 18
52	51	eleq2d 2687	. . . . . . . . . . . . . . . . 17
53	52	ad2antlr 763	. . . . . . . . . . . . . . . 16 Word Vtx $/\$ $/\$ $/\$
54	50, 53	mpbird 247	. . . . . . . . . . . . . . 15 Word Vtx $/\$ $/\$ $/\$
55		swrd0len 13422	. . . . . . . . . . . . . . 15 Word Vtx $/\$ substr
56	7, 54, 55	syl2anc 693	. . . . . . . . . . . . . 14 Word Vtx $/\$ $/\$ $/\$ substr
57	56	oveq1d 6665	. . . . . . . . . . . . 13 Word Vtx $/\$ $/\$ $/\$ substr
58	57	oveq2d 6666	. . . . . . . . . . . 12 Word Vtx $/\$ $/\$ $/\$ ..^ substr ..^
59	58	raleqdv 3144	. . . . . . . . . . 11 Word Vtx $/\$ $/\$ $/\$ ..^ substr substr substr Edg ..^ substr substr Edg
60	7	adantr 481	. . . . . . . . . . . . . . 15 Word Vtx $/\$ $/\$ $/\$ $/\$ ..^ Word Vtx
61	54	adantr 481	. . . . . . . . . . . . . . 15 Word Vtx $/\$ $/\$ $/\$ $/\$ ..^
62	30	ad2antrl 764	. . . . . . . . . . . . . . . . 17 Word Vtx $/\$ $/\$ $/\$
63		fzoss2 12496	. . . . . . . . . . . . . . . . 17 ..^ ..^
64	62, 63	syl 17	. . . . . . . . . . . . . . . 16 Word Vtx $/\$ $/\$ $/\$ ..^ ..^
65	64	sselda 3603	. . . . . . . . . . . . . . 15 Word Vtx $/\$ $/\$ $/\$ $/\$ ..^ ..^
66		swrd0fv 13439	. . . . . . . . . . . . . . 15 Word Vtx $/\$ $/\$ ..^ substr
67	60, 61, 65, 66	syl3anc 1326	. . . . . . . . . . . . . 14 Word Vtx $/\$ $/\$ $/\$ $/\$ ..^ substr
68	27	ad2antrl 764	. . . . . . . . . . . . . . . 16 Word Vtx $/\$ $/\$ $/\$
69		elfzom1elp1fzo 12534	. . . . . . . . . . . . . . . 16 $/\$ ..^ ..^
70	68, 69	sylan 488	. . . . . . . . . . . . . . 15 Word Vtx $/\$ $/\$ $/\$ $/\$ ..^ ..^
71		swrd0fv 13439	. . . . . . . . . . . . . . 15 Word Vtx $/\$ $/\$ ..^ substr
72	60, 61, 70, 71	syl3anc 1326	. . . . . . . . . . . . . 14 Word Vtx $/\$ $/\$ $/\$ $/\$ ..^ substr
73	67, 72	preq12d 4276	. . . . . . . . . . . . 13 Word Vtx $/\$ $/\$ $/\$ $/\$ ..^ substr substr
74	73	eleq1d 2686	. . . . . . . . . . . 12 Word Vtx $/\$ $/\$ $/\$ $/\$ ..^ substr substr Edg Edg
75	74	ralbidva 2985	. . . . . . . . . . 11 Word Vtx $/\$ $/\$ $/\$ ..^ substr substr Edg ..^ Edg
76	59, 75	bitrd 268	. . . . . . . . . 10 Word Vtx $/\$ $/\$ $/\$ ..^ substr substr substr Edg ..^ Edg
77	76	3adantl3 1219	. . . . . . . . 9 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ ..^ substr substr substr Edg ..^ Edg
78	77	adantr 481	. . . . . . . 8 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ $/\$ ..^ substr substr substr Edg ..^ Edg
79	42, 78	mpbird 247	. . . . . . 7 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ $/\$ ..^ substr substr substr Edg
80		elfz1uz 12410	. . . . . . . . . . . . . 14 $/\$
81		fzelp1 12393	. . . . . . . . . . . . . 14
82	80, 81	syl 17	. . . . . . . . . . . . 13 $/\$
83	82	adantl 482	. . . . . . . . . . . 12 Word Vtx $/\$ $/\$ $/\$
84		oveq2 6658	. . . . . . . . . . . . . 14
85	84	eleq2d 2687	. . . . . . . . . . . . 13
86	85	ad2antlr 763	. . . . . . . . . . . 12 Word Vtx $/\$ $/\$ $/\$
87	83, 86	mpbird 247	. . . . . . . . . . 11 Word Vtx $/\$ $/\$ $/\$
88		swrd0fvlsw 13443	. . . . . . . . . . . 12 Word Vtx $/\$ lastS substr
89		swrd0fv0 13440	. . . . . . . . . . . 12 Word Vtx $/\$ substr
90	88, 89	preq12d 4276	. . . . . . . . . . 11 Word Vtx $/\$ lastS substr substr
91	7, 87, 90	syl2anc 693	. . . . . . . . . 10 Word Vtx $/\$ $/\$ $/\$ lastS substr substr
92	91	3adantl3 1219	. . . . . . . . 9 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ lastS substr substr
93	92	adantr 481	. . . . . . . 8 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ $/\$ lastS substr substr
94		fz1fzo0m1 12515	. . . . . . . . . . . . . . . 16 ..^
95	80, 94	syl 17	. . . . . . . . . . . . . . 15 $/\$ ..^
96	95	3ad2ant3 1084	. . . . . . . . . . . . . 14 Word Vtx $/\$ $/\$ $/\$ ..^
97		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 $/\$
98	97	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 $/\$
99		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . 22
100		nncn 11028	. . . . . . . . . . . . . . . . . . . . . . 23
101		npcan1 10455	. . . . . . . . . . . . . . . . . . . . . . 23
102	100, 101	syl 17	. . . . . . . . . . . . . . . . . . . . . 22
103	99, 102	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . . . 21 $/\$
104	103	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 $/\$
105	98, 104	preq12d 4276	. . . . . . . . . . . . . . . . . . 19 $/\$
106	105	eleq1d 2686	. . . . . . . . . . . . . . . . . 18 $/\$ Edg Edg
107	106	ex 450	. . . . . . . . . . . . . . . . 17 Edg Edg
108	107	adantr 481	. . . . . . . . . . . . . . . 16 $/\$ Edg Edg
109	108	3ad2ant3 1084	. . . . . . . . . . . . . . 15 Word Vtx $/\$ $/\$ $/\$ Edg Edg
110	109	imp 445	. . . . . . . . . . . . . 14 Word Vtx $/\$ $/\$ $/\$ $/\$ Edg Edg
111	96, 110	rspcdv 3312	. . . . . . . . . . . . 13 Word Vtx $/\$ $/\$ $/\$ ..^ Edg Edg
112	111	3exp 1264	. . . . . . . . . . . 12 Word Vtx $/\$ ..^ Edg Edg
113	112	com34 91	. . . . . . . . . . 11 Word Vtx ..^ Edg $/\$ Edg
114	113	3imp1 1280	. . . . . . . . . 10 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ Edg
115	114	adantr 481	. . . . . . . . 9 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ $/\$ Edg
116		preq2 4269	. . . . . . . . . . 11
117	116	eleq1d 2686	. . . . . . . . . 10 Edg Edg
118	117	adantl 482	. . . . . . . . 9 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ $/\$ Edg Edg
119	115, 118	mpbid 222	. . . . . . . 8 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ $/\$ Edg
120	93, 119	eqeltrd 2701	. . . . . . 7 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ $/\$ lastS substr substr Edg
121	26, 79, 120	3jca 1242	. . . . . 6 Word Vtx $/\$ $/\$ ..^ Edg $/\$ $/\$ $/\$ substr Word Vtx $/\$ substr $/\$ ..^ substr substr substr Edg $/\$ lastS substr substr Edg
122	121	exp31 630	. . . . 5 Word Vtx $/\$ $/\$ ..^ Edg $/\$ substr Word Vtx $/\$ substr $/\$ ..^ substr substr substr Edg $/\$ lastS substr substr Edg
123	122	3imp21 1277	. . . 4 $/\$ $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ substr Word Vtx $/\$ substr $/\$ ..^ substr substr substr Edg $/\$ lastS substr substr Edg
124	1, 2	isclwwlks 26880	. . . 4 substr ClWWalks substr Word Vtx $/\$ substr $/\$ ..^ substr substr substr Edg $/\$ lastS substr substr Edg
125	123, 124	sylibr 224	. . 3 $/\$ $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ substr ClWWalks
126	47	adantl 482	. . . . . . . . . . 11 Word Vtx $/\$ $/\$ $/\$
127	126, 48	syl 17	. . . . . . . . . 10 Word Vtx $/\$ $/\$ $/\$
128	127, 53	mpbird 247	. . . . . . . . 9 Word Vtx $/\$ $/\$ $/\$
129	7, 128	jca 554	. . . . . . . 8 Word Vtx $/\$ $/\$ $/\$ Word Vtx $/\$
130	129	ex 450	. . . . . . 7 Word Vtx $/\$ $/\$ Word Vtx $/\$
131	130	3adant3 1081	. . . . . 6 Word Vtx $/\$ $/\$ ..^ Edg $/\$ Word Vtx $/\$
132	131	impcom 446	. . . . 5 $/\$ $/\$ Word Vtx $/\$ $/\$ ..^ Edg Word Vtx $/\$
133	132	3adant3 1081	. . . 4 $/\$ $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ Word Vtx $/\$
134	133, 55	syl 17	. . 3 $/\$ $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ substr
135		isclwwlksn 26882	. . . . 5 substr ClWWalksN substr ClWWalks $/\$ substr
136	135	adantr 481	. . . 4 $/\$ substr ClWWalksN substr ClWWalks $/\$ substr
137	136	3ad2ant1 1082	. . 3 $/\$ $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ substr ClWWalksN substr ClWWalks $/\$ substr
138	125, 134, 137	mpbir2and 957	. 2 $/\$ $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ substr ClWWalksN
139	3, 138	syl3an2 1360	1 $/\$ $/\$ WWalksN $/\$ substr ClWWalksN

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912

wss 3574

c0 3915

cpr 4179

cop 4183 class class class wbr 4653

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cle 10075

cmin 10266

cn 11020

cn0 11292

cz 11377

cuz 11687

cfz 12326 ..^cfzo 12465

chash 13117 Word cword 13291 lastS clsw 13292 substr csubstr 13295 Vtxcvtx 25874 Edgcedg 25939 WWalksN cwwlksn 26718 ClWWalkscclwwlks 26875 ClWWalksN cclwwlksn 26876

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 df-clwwlks 26877 df-clwwlksn 26878

This theorem is referenced by: extwwlkfablem2 27210

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