Theorem clwlksf1clwwlk 26969

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 26962	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺))
6		simprl 794	. . . . . 6 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → 𝑢 ∈ 𝐶)
7		ovex 6678	. . . . . 6 ⊢ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) ∈ V
8		fveq2 6191	. . . . . . . . 9 ⊢ (𝑐 = 𝑢 → (2nd ‘𝑐) = (2nd ‘𝑢))
9	2, 8	syl5eq 2668	. . . . . . . 8 ⊢ (𝑐 = 𝑢 → 𝐵 = (2nd ‘𝑢))
10		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑢 → (1st ‘𝑐) = (1st ‘𝑢))
11	1, 10	syl5eq 2668	. . . . . . . . . 10 ⊢ (𝑐 = 𝑢 → 𝐴 = (1st ‘𝑢))
12	11	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑐 = 𝑢 → (#‘𝐴) = (#‘(1st ‘𝑢)))
13	12	opeq2d 4409	. . . . . . . 8 ⊢ (𝑐 = 𝑢 → ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st ‘𝑢))⟩)
14	9, 13	oveq12d 6668	. . . . . . 7 ⊢ (𝑐 = 𝑢 → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩))
15	14, 4	fvmptg 6280	. . . . . 6 ⊢ ((𝑢 ∈ 𝐶 ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) ∈ V) → (𝐹‘𝑢) = ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩))
16	6, 7, 15	sylancl 694	. . . . 5 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → (𝐹‘𝑢) = ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩))
17		simpr 477	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → 𝑤 ∈ 𝐶)
18		ovex 6678	. . . . . . . 8 ⊢ ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) ∈ V
19	17, 18	jctir 561	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (𝑤 ∈ 𝐶 ∧ ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) ∈ V))
20	19	adantl 482	. . . . . 6 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → (𝑤 ∈ 𝐶 ∧ ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) ∈ V))
21		fveq2 6191	. . . . . . . . 9 ⊢ (𝑐 = 𝑤 → (2nd ‘𝑐) = (2nd ‘𝑤))
22	2, 21	syl5eq 2668	. . . . . . . 8 ⊢ (𝑐 = 𝑤 → 𝐵 = (2nd ‘𝑤))
23		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑤 → (1st ‘𝑐) = (1st ‘𝑤))
24	1, 23	syl5eq 2668	. . . . . . . . . 10 ⊢ (𝑐 = 𝑤 → 𝐴 = (1st ‘𝑤))
25	24	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑐 = 𝑤 → (#‘𝐴) = (#‘(1st ‘𝑤)))
26	25	opeq2d 4409	. . . . . . . 8 ⊢ (𝑐 = 𝑤 → ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st ‘𝑤))⟩)
27	22, 26	oveq12d 6668	. . . . . . 7 ⊢ (𝑐 = 𝑤 → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩))
28	27, 4	fvmptg 6280	. . . . . 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 2637	. . . 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 26959	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐶 → (#‘(1st ‘𝑤)) = 𝑁)
32	31	eqcomd 2628	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → 𝑁 = (#‘(1st ‘𝑤)))
33	32	adantl 482	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → 𝑁 = (#‘(1st ‘𝑤)))
34	33	ad2antlr 763	. . . . . 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 15388	. . . . . . . . 9 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
36	35	ad2antlr 763	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → 𝑁 ∈ ℕ)
37	17	adantl 482	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → 𝑤 ∈ 𝐶)
38	1, 2, 3, 4	clwlksf1clwwlklem 26968	. . . . . . . 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 1326	. . . . . . 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 445	. . . . . 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 26214	. . . . . . . . 9 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
42		usgruspgr 26073	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
43	41, 42	syl 17	. . . . . . . 8 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USPGraph )
44	43	ad3antrrr 766	. . . . . . 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 3359	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑢 ∈ (ClWalks‘𝐺))
46		clwlkwlk 26671	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (ClWalks‘𝐺) → 𝑢 ∈ (Walks‘𝐺))
47	45, 46	syl 17	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑢 ∈ (Walks‘𝐺))
48	47, 3	eleq2s 2719	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐶 → 𝑢 ∈ (Walks‘𝐺))
49		elrabi 3359	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑤 ∈ (ClWalks‘𝐺))
50		clwlkwlk 26671	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑤 ∈ (Walks‘𝐺))
52	51, 3	eleq2s 2719	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐶 → 𝑤 ∈ (Walks‘𝐺))
53	48, 52	anim12i 590	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (𝑢 ∈ (Walks‘𝐺) ∧ 𝑤 ∈ (Walks‘𝐺)))
54	53	ad2antlr 763	. . . . . . 7 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → (𝑢 ∈ (Walks‘𝐺) ∧ 𝑤 ∈ (Walks‘𝐺)))
55	1, 2, 3, 4	clwlksfclwwlk1hashn 26959	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝐶 → (#‘(1st ‘𝑢)) = 𝑁)
56	55	eqcomd 2628	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐶 → 𝑁 = (#‘(1st ‘𝑢)))
57	56	adantr 481	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → 𝑁 = (#‘(1st ‘𝑢)))
58	57	ad2antlr 763	. . . . . . 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 26542	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑢 ∈ (Walks‘𝐺) ∧ 𝑤 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝑢))) → (𝑢 = 𝑤 ↔ (𝑁 = (#‘(1st ‘𝑤)) ∧ ∀𝑖 ∈ (0...𝑁)((2nd ‘𝑢)‘𝑖) = ((2nd ‘𝑤)‘𝑖))))
60	44, 54, 58, 59	syl3anc 1326	. . . . . 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 957	. . . . 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 450	. . . 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 230	. . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → ((𝐹‘𝑢) = (𝐹‘𝑤) → 𝑢 = 𝑤))
64	63	ralrimivva 2971	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ∀𝑢 ∈ 𝐶 ∀𝑤 ∈ 𝐶 ((𝐹‘𝑢) = (𝐹‘𝑤) → 𝑢 = 𝑤))
65		dff13 6512	. 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 698	1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶–1-1→(𝑁 ClWWalksN 𝐺))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200  ⟨cop 4183   ↦ cmpt 4729  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  0cc0 9936  ℕcn 11020  ...cfz 12326  #chash 13117   substr csubstr 13295  ℙcprime 15385   USPGraph cuspgr 26043   USGraph cusgr 26044   FinUSGraph cfusgr 26208  Walkscwlks 26492  ClWalkscclwlks 26666   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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  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-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-sup 8348  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-substr 13303  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-wlks 26495  df-clwlks 26667  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  clwlksf1oclwwlk  26970