Theorem clwlksf1clwwlk 41276

Description: There is a one-to-one function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.)

Hypotheses

Ref	Expression
clwlksfclwwlk.1	⊢ 𝐴 = (1st ‘𝑐)
clwlksfclwwlk.2	⊢ 𝐵 = (2nd ‘𝑐)
clwlksfclwwlk.c	⊢ 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
clwlksfclwwlk.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))

Assertion

Ref	Expression
clwlksf1clwwlk	⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶–1-1→(𝑁 ClWWalkSN 𝐺))

Distinct variable groups:   𝐺,𝑐   𝑁,𝑐   𝐶,𝑐   𝐹,𝑐

Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlksf1clwwlk

Dummy variables 𝑖 𝑤 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlksfclwwlk.1	. . 3 ⊢ 𝐴 = (1st ‘𝑐)
2		clwlksfclwwlk.2	. . 3 ⊢ 𝐵 = (2nd ‘𝑐)
3		clwlksfclwwlk.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
4		clwlksfclwwlk.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
5	1, 2, 3, 4	clwlksfclwwlk 41269	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalkSN 𝐺))
6		simprl 790	. . . . . 6 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → 𝑢 ∈ 𝐶)
7		ovex 6577	. . . . . 6 ⊢ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) ∈ V
8		fveq2 6103	. . . . . . . . 9 ⊢ (𝑐 = 𝑢 → (2nd ‘𝑐) = (2nd ‘𝑢))
9	2, 8	syl5eq 2656	. . . . . . . 8 ⊢ (𝑐 = 𝑢 → 𝐵 = (2nd ‘𝑢))
10		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑢 → (1st ‘𝑐) = (1st ‘𝑢))
11	1, 10	syl5eq 2656	. . . . . . . . . 10 ⊢ (𝑐 = 𝑢 → 𝐴 = (1st ‘𝑢))
12	11	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑐 = 𝑢 → (#‘𝐴) = (#‘(1st ‘𝑢)))
13	12	opeq2d 4347	. . . . . . . 8 ⊢ (𝑐 = 𝑢 → ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st ‘𝑢))⟩)
14	9, 13	oveq12d 6567	. . . . . . 7 ⊢ (𝑐 = 𝑢 → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩))
15	14, 4	fvmptg 6189	. . . . . 6 ⊢ ((𝑢 ∈ 𝐶 ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) ∈ V) → (𝐹‘𝑢) = ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩))
16	6, 7, 15	sylancl 693	. . . . 5 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → (𝐹‘𝑢) = ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩))
17		simpr 476	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → 𝑤 ∈ 𝐶)
18		ovex 6577	. . . . . . . 8 ⊢ ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) ∈ V
19	17, 18	jctir 559	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (𝑤 ∈ 𝐶 ∧ ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) ∈ V))
20	19	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → (𝑤 ∈ 𝐶 ∧ ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) ∈ V))
21		fveq2 6103	. . . . . . . . 9 ⊢ (𝑐 = 𝑤 → (2nd ‘𝑐) = (2nd ‘𝑤))
22	2, 21	syl5eq 2656	. . . . . . . 8 ⊢ (𝑐 = 𝑤 → 𝐵 = (2nd ‘𝑤))
23		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑤 → (1st ‘𝑐) = (1st ‘𝑤))
24	1, 23	syl5eq 2656	. . . . . . . . . 10 ⊢ (𝑐 = 𝑤 → 𝐴 = (1st ‘𝑤))
25	24	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑐 = 𝑤 → (#‘𝐴) = (#‘(1st ‘𝑤)))
26	25	opeq2d 4347	. . . . . . . 8 ⊢ (𝑐 = 𝑤 → ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st ‘𝑤))⟩)
27	22, 26	oveq12d 6567	. . . . . . 7 ⊢ (𝑐 = 𝑤 → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩))
28	27, 4	fvmptg 6189	. . . . . 6 ⊢ ((𝑤 ∈ 𝐶 ∧ ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) ∈ V) → (𝐹‘𝑤) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩))
29	20, 28	syl 17	. . . . 5 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → (𝐹‘𝑤) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩))
30	16, 29	eqeq12d 2625	. . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → ((𝐹‘𝑢) = (𝐹‘𝑤) ↔ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)))
31	1, 2, 3, 4	clwlksfclwwlk1hashn 41266	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐶 → (#‘(1st ‘𝑤)) = 𝑁)
32	31	eqcomd 2616	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → 𝑁 = (#‘(1st ‘𝑤)))
33	32	adantl 481	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → 𝑁 = (#‘(1st ‘𝑤)))
34	33	ad2antlr 759	. . . . . 6 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → 𝑁 = (#‘(1st ‘𝑤)))
35		prmnn 15226	. . . . . . . . 9 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
36	35	ad2antlr 759	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → 𝑁 ∈ ℕ)
37	17	adantl 481	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → 𝑤 ∈ 𝐶)
38	1, 2, 3, 4	clwlksf1clwwlklem 41275	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) → ∀𝑖 ∈ (0...𝑁)((2nd ‘𝑢)‘𝑖) = ((2nd ‘𝑤)‘𝑖)))
39	36, 6, 37, 38	syl3anc 1318	. . . . . . 7 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → (((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) → ∀𝑖 ∈ (0...𝑁)((2nd ‘𝑢)‘𝑖) = ((2nd ‘𝑤)‘𝑖)))
40	39	imp 444	. . . . . 6 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → ∀𝑖 ∈ (0...𝑁)((2nd ‘𝑢)‘𝑖) = ((2nd ‘𝑤)‘𝑖))
41		fusgrusgr 40541	. . . . . . . . 9 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
42		usgruspgr 40408	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
43	41, 42	syl 17	. . . . . . . 8 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USPGraph )
44	43	ad3antrrr 762	. . . . . . 7 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → 𝐺 ∈ USPGraph )
45		elrabi 3328	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑢 ∈ (ClWalkS‘𝐺))
46		clwlk1wlk 40982	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (ClWalkS‘𝐺) → 𝑢 ∈ (1Walks‘𝐺))
47	45, 46	syl 17	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑢 ∈ (1Walks‘𝐺))
48	47, 3	eleq2s 2706	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐶 → 𝑢 ∈ (1Walks‘𝐺))
49		elrabi 3328	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑤 ∈ (ClWalkS‘𝐺))
50		clwlk1wlk 40982	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (ClWalkS‘𝐺) → 𝑤 ∈ (1Walks‘𝐺))
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑤 ∈ (1Walks‘𝐺))
52	51, 3	eleq2s 2706	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐶 → 𝑤 ∈ (1Walks‘𝐺))
53	48, 52	anim12i 588	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (𝑢 ∈ (1Walks‘𝐺) ∧ 𝑤 ∈ (1Walks‘𝐺)))
54	53	ad2antlr 759	. . . . . . 7 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → (𝑢 ∈ (1Walks‘𝐺) ∧ 𝑤 ∈ (1Walks‘𝐺)))
55	1, 2, 3, 4	clwlksfclwwlk1hashn 41266	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝐶 → (#‘(1st ‘𝑢)) = 𝑁)
56	55	eqcomd 2616	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐶 → 𝑁 = (#‘(1st ‘𝑢)))
57	56	adantr 480	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → 𝑁 = (#‘(1st ‘𝑢)))
58	57	ad2antlr 759	. . . . . . 7 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → 𝑁 = (#‘(1st ‘𝑢)))
59		uspgr2wlkeq 40854	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑢 ∈ (1Walks‘𝐺) ∧ 𝑤 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝑢))) → (𝑢 = 𝑤 ↔ (𝑁 = (#‘(1st ‘𝑤)) ∧ ∀𝑖 ∈ (0...𝑁)((2nd ‘𝑢)‘𝑖) = ((2nd ‘𝑤)‘𝑖))))
60	44, 54, 58, 59	syl3anc 1318	. . . . . 6 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → (𝑢 = 𝑤 ↔ (𝑁 = (#‘(1st ‘𝑤)) ∧ ∀𝑖 ∈ (0...𝑁)((2nd ‘𝑢)‘𝑖) = ((2nd ‘𝑤)‘𝑖))))
61	34, 40, 60	mpbir2and 959	. . . . 5 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → 𝑢 = 𝑤)
62	61	ex 449	. . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → (((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) → 𝑢 = 𝑤))
63	30, 62	sylbid 229	. . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → ((𝐹‘𝑢) = (𝐹‘𝑤) → 𝑢 = 𝑤))
64	63	ralrimivva 2954	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ∀𝑢 ∈ 𝐶 ∀𝑤 ∈ 𝐶 ((𝐹‘𝑢) = (𝐹‘𝑤) → 𝑢 = 𝑤))
65		dff13 6416	. 2 ⊢ (𝐹:𝐶–1-1→(𝑁 ClWWalkSN 𝐺) ↔ (𝐹:𝐶⟶(𝑁 ClWWalkSN 𝐺) ∧ ∀𝑢 ∈ 𝐶 ∀𝑤 ∈ 𝐶 ((𝐹‘𝑢) = (𝐹‘𝑤) → 𝑢 = 𝑤)))
66	5, 64, 65	sylanbrc 695	1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶–1-1→(𝑁 ClWWalkSN 𝐺))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  0cc0 9815  ℕcn 10897  ...cfz 12197  #chash 12979   substr csubstr 13150  ℙcprime 15223   USPGraph cuspgr 40378   USGraph cusgr 40379   FinUSGraph cfusgr 40535  1Walksc1wlks 40796  ClWalkScclwlks 40976   ClWWalkSN cclwwlksn 41184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-substr 13158  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-prm 15224  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-1wlks 40800  df-wlks 40801  df-clwlks 40977  df-clwwlks 41185  df-clwwlksn 41186

This theorem is referenced by:  clwlksf1oclwwlk  41277