clwwlksel - Metamath Proof Explorer

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Theorem clwwlksel 26914

Description: Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 25-Apr-2021.)

**Hypothesis**
Ref	Expression
clwwlksbij.d	WWalksN lastS

**Assertion**
Ref	Expression
clwwlksel	$/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++

Distinct variable groups:

Allowed substitution hints:

(

)

**Proof of Theorem clwwlksel**
Step	Hyp	Ref	Expression
1		simprl 794	. . . . . 6 $/\$ Word Vtx $/\$ Word Vtx
2		fstwrdne0 13345	. . . . . 6 $/\$ Word Vtx $/\$ Vtx
3		ccatws1n0 13409	. . . . . 6 Word Vtx $/\$ Vtx ++
4	1, 2, 3	syl2anc 693	. . . . 5 $/\$ Word Vtx $/\$ ++
5	4	3adant3 1081	. . . 4 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++
6		simp2l 1087	. . . . 5 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg Word Vtx
7	2	s1cld 13383	. . . . . 6 $/\$ Word Vtx $/\$ Word Vtx
8	7	3adant3 1081	. . . . 5 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg Word Vtx
9		ccatcl 13359	. . . . 5 Word Vtx $/\$ Word Vtx ++ Word Vtx
10	6, 8, 9	syl2anc 693	. . . 4 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++ Word Vtx
11	1	adantr 481	. . . . . . . . . . . . . . 15 $/\$ Word Vtx $/\$ $/\$ ..^ Word Vtx
12	7	adantr 481	. . . . . . . . . . . . . . 15 $/\$ Word Vtx $/\$ $/\$ ..^ Word Vtx
13		elfzonn0 12512	. . . . . . . . . . . . . . . . . . 19 ..^
14	13	adantl 482	. . . . . . . . . . . . . . . . . 18 $/\$ ..^
15		nnz 11399	. . . . . . . . . . . . . . . . . . 19
16	15	adantr 481	. . . . . . . . . . . . . . . . . 18 $/\$ ..^
17		elfzo0 12508	. . . . . . . . . . . . . . . . . . . 20 ..^ $/\$ $/\$
18		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . . 27
19	18	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 $/\$
20		nnre 11027	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
21		peano2rem 10348	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
22	20, 21	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27
23	22	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 $/\$
24	20	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 $/\$
25	19, 23, 24	3jca 1242	. . . . . . . . . . . . . . . . . . . . . . . . 25 $/\$ $/\$ $/\$
26	25	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$
27		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$
28	20	ltm1d 10956	. . . . . . . . . . . . . . . . . . . . . . . . . 26
29	28	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 $/\$
30	29	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$
31		lttr 10114	. . . . . . . . . . . . . . . . . . . . . . . . 25 $/\$ $/\$ $/\$
32	31	imp 445	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$
33	26, 27, 30, 32	syl12anc 1324	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$
34	33	ex 450	. . . . . . . . . . . . . . . . . . . . . 22 $/\$
35	34	impancom 456	. . . . . . . . . . . . . . . . . . . . 21 $/\$
36	35	3adant2 1080	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
37	17, 36	sylbi 207	. . . . . . . . . . . . . . . . . . 19 ..^
38	37	impcom 446	. . . . . . . . . . . . . . . . . 18 $/\$ ..^
39		elfzo0z 12509	. . . . . . . . . . . . . . . . . 18 ..^ $/\$ $/\$
40	14, 16, 38, 39	syl3anbrc 1246	. . . . . . . . . . . . . . . . 17 $/\$ ..^ ..^
41	40	adantlr 751	. . . . . . . . . . . . . . . 16 $/\$ Word Vtx $/\$ $/\$ ..^ ..^
42		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 ..^ ..^
43	42	eleq2d 2687	. . . . . . . . . . . . . . . . . 18 ..^ ..^
44	43	ad2antll 765	. . . . . . . . . . . . . . . . 17 $/\$ Word Vtx $/\$ ..^ ..^
45	44	adantr 481	. . . . . . . . . . . . . . . 16 $/\$ Word Vtx $/\$ $/\$ ..^ ..^ ..^
46	41, 45	mpbird 247	. . . . . . . . . . . . . . 15 $/\$ Word Vtx $/\$ $/\$ ..^ ..^
47		ccatval1 13361	. . . . . . . . . . . . . . 15 Word Vtx $/\$ Word Vtx $/\$ ..^ ++
48	11, 12, 46, 47	syl3anc 1326	. . . . . . . . . . . . . 14 $/\$ Word Vtx $/\$ $/\$ ..^ ++
49	48	eqcomd 2628	. . . . . . . . . . . . 13 $/\$ Word Vtx $/\$ $/\$ ..^ ++
50		elfzom1p1elfzo 12547	. . . . . . . . . . . . . . . . 17 $/\$ ..^ ..^
51	50	adantlr 751	. . . . . . . . . . . . . . . 16 $/\$ Word Vtx $/\$ $/\$ ..^ ..^
52	42	ad2antll 765	. . . . . . . . . . . . . . . . 17 $/\$ Word Vtx $/\$ ..^ ..^
53	52	adantr 481	. . . . . . . . . . . . . . . 16 $/\$ Word Vtx $/\$ $/\$ ..^ ..^ ..^
54	51, 53	eleqtrrd 2704	. . . . . . . . . . . . . . 15 $/\$ Word Vtx $/\$ $/\$ ..^ ..^
55		ccatval1 13361	. . . . . . . . . . . . . . 15 Word Vtx $/\$ Word Vtx $/\$ ..^ ++
56	11, 12, 54, 55	syl3anc 1326	. . . . . . . . . . . . . 14 $/\$ Word Vtx $/\$ $/\$ ..^ ++
57	56	eqcomd 2628	. . . . . . . . . . . . 13 $/\$ Word Vtx $/\$ $/\$ ..^ ++
58	49, 57	preq12d 4276	. . . . . . . . . . . 12 $/\$ Word Vtx $/\$ $/\$ ..^ ++ ++
59	58	eleq1d 2686	. . . . . . . . . . 11 $/\$ Word Vtx $/\$ $/\$ ..^ Edg ++ ++ Edg
60	59	ralbidva 2985	. . . . . . . . . 10 $/\$ Word Vtx $/\$ ..^ Edg ..^ ++ ++ Edg
61	60	biimpcd 239	. . . . . . . . 9 ..^ Edg $/\$ Word Vtx $/\$ ..^ ++ ++ Edg
62	61	adantr 481	. . . . . . . 8 ..^ Edg $/\$ lastS Edg $/\$ Word Vtx $/\$ ..^ ++ ++ Edg
63	62	expdcom 455	. . . . . . 7 Word Vtx $/\$ ..^ Edg $/\$ lastS Edg ..^ ++ ++ Edg
64	63	3imp 1256	. . . . . 6 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ..^ ++ ++ Edg
65		fzo0end 12560	. . . . . . . . . . . . . . . . 17 ..^
66	65	adantr 481	. . . . . . . . . . . . . . . 16 $/\$ Word Vtx $/\$ ..^
67	42	eleq2d 2687	. . . . . . . . . . . . . . . . 17 ..^ ..^
68	67	ad2antll 765	. . . . . . . . . . . . . . . 16 $/\$ Word Vtx $/\$ ..^ ..^
69	66, 68	mpbird 247	. . . . . . . . . . . . . . 15 $/\$ Word Vtx $/\$ ..^
70		ccatval1 13361	. . . . . . . . . . . . . . 15 Word Vtx $/\$ Word Vtx $/\$ ..^ ++
71	1, 7, 69, 70	syl3anc 1326	. . . . . . . . . . . . . 14 $/\$ Word Vtx $/\$ ++
72		oveq1 6657	. . . . . . . . . . . . . . . . . . 19
73	72	fveq2d 6195	. . . . . . . . . . . . . . . . . 18
74	73	eqcoms 2630	. . . . . . . . . . . . . . . . 17
75	74	adantl 482	. . . . . . . . . . . . . . . 16 Word Vtx $/\$
76		lsw 13351	. . . . . . . . . . . . . . . . 17 Word Vtx lastS
77	76	adantr 481	. . . . . . . . . . . . . . . 16 Word Vtx $/\$ lastS
78	75, 77	eqtr4d 2659	. . . . . . . . . . . . . . 15 Word Vtx $/\$ lastS
79	78	adantl 482	. . . . . . . . . . . . . 14 $/\$ Word Vtx $/\$ lastS
80	71, 79	eqtr2d 2657	. . . . . . . . . . . . 13 $/\$ Word Vtx $/\$ lastS ++
81		fveq2 6191	. . . . . . . . . . . . . . . 16 ++ ++
82	81	eqcoms 2630	. . . . . . . . . . . . . . 15 ++ ++
83	82	ad2antll 765	. . . . . . . . . . . . . 14 $/\$ Word Vtx $/\$ ++ ++
84		nncn 11028	. . . . . . . . . . . . . . . . . 18
85		1cnd 10056	. . . . . . . . . . . . . . . . . 18
86	84, 85	npcand 10396	. . . . . . . . . . . . . . . . 17
87	86	eqcomd 2628	. . . . . . . . . . . . . . . 16
88	87	fveq2d 6195	. . . . . . . . . . . . . . 15 ++ ++
89	88	adantr 481	. . . . . . . . . . . . . 14 $/\$ Word Vtx $/\$ ++ ++
90		ccatws1ls 13410	. . . . . . . . . . . . . . 15 Word Vtx $/\$ Vtx ++
91	1, 2, 90	syl2anc 693	. . . . . . . . . . . . . 14 $/\$ Word Vtx $/\$ ++
92	83, 89, 91	3eqtr3rd 2665	. . . . . . . . . . . . 13 $/\$ Word Vtx $/\$ ++
93	80, 92	preq12d 4276	. . . . . . . . . . . 12 $/\$ Word Vtx $/\$ lastS ++ ++
94	93	eleq1d 2686	. . . . . . . . . . 11 $/\$ Word Vtx $/\$ lastS Edg ++ ++ Edg
95	94	biimpcd 239	. . . . . . . . . 10 lastS Edg $/\$ Word Vtx $/\$ ++ ++ Edg
96	95	adantl 482	. . . . . . . . 9 ..^ Edg $/\$ lastS Edg $/\$ Word Vtx $/\$ ++ ++ Edg
97	96	expdcom 455	. . . . . . . 8 Word Vtx $/\$ ..^ Edg $/\$ lastS Edg ++ ++ Edg
98	97	3imp 1256	. . . . . . 7 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++ ++ Edg
99		ovex 6678	. . . . . . . 8
100		fveq2 6191	. . . . . . . . . 10 ++ ++
101		oveq1 6657	. . . . . . . . . . 11
102	101	fveq2d 6195	. . . . . . . . . 10 ++ ++
103	100, 102	preq12d 4276	. . . . . . . . 9 ++ ++ ++ ++
104	103	eleq1d 2686	. . . . . . . 8 ++ ++ Edg ++ ++ Edg
105	99, 104	ralsn 4222	. . . . . . 7 ++ ++ Edg ++ ++ Edg
106	98, 105	sylibr 224	. . . . . 6 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++ ++ Edg
107	84, 85, 85	addsubd 10413	. . . . . . . . . . 11
108	107	oveq2d 6666	. . . . . . . . . 10 ..^ ..^
109		nnm1nn0 11334	. . . . . . . . . . . 12
110		elnn0uz 11725	. . . . . . . . . . . 12
111	109, 110	sylib 208	. . . . . . . . . . 11
112		fzosplitsn 12576	. . . . . . . . . . 11 ..^ ..^
113	111, 112	syl 17	. . . . . . . . . 10 ..^ ..^
114	108, 113	eqtrd 2656	. . . . . . . . 9 ..^ ..^
115	114	raleqdv 3144	. . . . . . . 8 ..^ ++ ++ Edg ..^ ++ ++ Edg
116		ralunb 3794	. . . . . . . 8 ..^ ++ ++ Edg ..^ ++ ++ Edg $/\$ ++ ++ Edg
117	115, 116	syl6bb 276	. . . . . . 7 ..^ ++ ++ Edg ..^ ++ ++ Edg $/\$ ++ ++ Edg
118	117	3ad2ant1 1082	. . . . . 6 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ..^ ++ ++ Edg ..^ ++ ++ Edg $/\$ ++ ++ Edg
119	64, 106, 118	mpbir2and 957	. . . . 5 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ..^ ++ ++ Edg
120		ccatlen 13360	. . . . . . . . . . 11 Word Vtx $/\$ Word Vtx ++
121	1, 7, 120	syl2anc 693	. . . . . . . . . 10 $/\$ Word Vtx $/\$ ++
122		id 22	. . . . . . . . . . . 12
123		s1len 13385	. . . . . . . . . . . . 13
124	123	a1i 11	. . . . . . . . . . . 12
125	122, 124	oveq12d 6668	. . . . . . . . . . 11
126	125	ad2antll 765	. . . . . . . . . 10 $/\$ Word Vtx $/\$
127	121, 126	eqtrd 2656	. . . . . . . . 9 $/\$ Word Vtx $/\$ ++
128	127	3adant3 1081	. . . . . . . 8 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++
129	128	oveq1d 6665	. . . . . . 7 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++
130	129	oveq2d 6666	. . . . . 6 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ..^ ++ ..^
131	130	raleqdv 3144	. . . . 5 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ..^ ++ ++ ++ Edg ..^ ++ ++ Edg
132	119, 131	mpbird 247	. . . 4 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ..^ ++ ++ ++ Edg
133	5, 10, 132	3jca 1242	. . 3 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++ $/\$ ++ Word Vtx $/\$ ..^ ++ ++ ++ Edg
134		nnnn0 11299	. . . . . 6
135		iswwlksn 26730	. . . . . 6 ++ WWalksN ++ WWalks $/\$ ++
136	134, 135	syl 17	. . . . 5 ++ WWalksN ++ WWalks $/\$ ++
137		eqid 2622	. . . . . . . 8 Vtx Vtx
138		eqid 2622	. . . . . . . 8 Edg Edg
139	137, 138	iswwlks 26728	. . . . . . 7 ++ WWalks ++ $/\$ ++ Word Vtx $/\$ ..^ ++ ++ ++ Edg
140	139	a1i 11	. . . . . 6 ++ WWalks ++ $/\$ ++ Word Vtx $/\$ ..^ ++ ++ ++ Edg
141	140	anbi1d 741	. . . . 5 ++ WWalks $/\$ ++ ++ $/\$ ++ Word Vtx $/\$ ..^ ++ ++ ++ Edg $/\$ ++
142	136, 141	bitrd 268	. . . 4 ++ WWalksN ++ $/\$ ++ Word Vtx $/\$ ..^ ++ ++ ++ Edg $/\$ ++
143	142	3ad2ant1 1082	. . 3 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++ WWalksN ++ $/\$ ++ Word Vtx $/\$ ..^ ++ ++ ++ Edg $/\$ ++
144	133, 128, 143	mpbir2and 957	. 2 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++ WWalksN
145		lswccats1 13411	. . . . 5 Word Vtx $/\$ Vtx lastS ++
146	1, 2, 145	syl2anc 693	. . . 4 $/\$ Word Vtx $/\$ lastS ++
147		lbfzo0 12507	. . . . . . . 8 ..^
148	147	biimpri 218	. . . . . . 7 ..^
149	148	adantr 481	. . . . . 6 $/\$ Word Vtx $/\$ ..^
150	42	eleq2d 2687	. . . . . . 7 ..^ ..^
151	150	ad2antll 765	. . . . . 6 $/\$ Word Vtx $/\$ ..^ ..^
152	149, 151	mpbird 247	. . . . 5 $/\$ Word Vtx $/\$ ..^
153		ccatval1 13361	. . . . 5 Word Vtx $/\$ Word Vtx $/\$ ..^ ++
154	1, 7, 152, 153	syl3anc 1326	. . . 4 $/\$ Word Vtx $/\$ ++
155	146, 154	eqtr4d 2659	. . 3 $/\$ Word Vtx $/\$ lastS ++ ++
156	155	3adant3 1081	. 2 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg lastS ++ ++
157		fveq2 6191	. . . 4 ++ lastS lastS ++
158		fveq1 6190	. . . 4 ++ ++
159	157, 158	eqeq12d 2637	. . 3 ++ lastS lastS ++ ++
160		clwwlksbij.d	. . 3 WWalksN lastS
161	159, 160	elrab2 3366	. 2 ++ ++ WWalksN $/\$ lastS ++ ++
162	144, 156, 161	sylanbrc 698	1 $/\$ Word Vtx $/\$ $/\$ ..^ Edg $/\$ lastS Edg ++

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990

wne 2794

wral 2912

crab 2916

cun 3572

c0 3915

csn 4177

cpr 4179 class class class wbr 4653

cfv 5888 (class class class)co 6650

cr 9935

cc0 9936

c1 9937

caddc 9939

clt 10074

cmin 10266

cn 11020

cn0 11292

cz 11377

cuz 11687 ..^cfzo 12465

chash 13117 Word cword 13291 lastS clsw 13292 ++ cconcat 13293

cs1 13294 Vtxcvtx 25874 Edgcedg 25939 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-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-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-wwlks 26722 df-wwlksn 26723

This theorem is referenced by: clwwlksfo 26918 clwwlksnwwlkncl 26921

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