Theorem clwlkf1clwwlk 26377

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.)

Hypotheses

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

Assertion

Ref	Expression
clwlkf1clwwlk	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶–1-1→((𝑉 ClWWalksN 𝐸)‘𝑁))

Distinct variable groups:   𝐸,𝑐   𝑁,𝑐   𝑉,𝑐   𝐶,𝑐   𝐹,𝑐

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

Proof of Theorem clwlkf1clwwlk

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

Step	Hyp	Ref	Expression
1		clwlkfclwwlk.1	. . 3 ⊢ 𝐴 = (1st ‘𝑐)
2		clwlkfclwwlk.2	. . 3 ⊢ 𝐵 = (2nd ‘𝑐)
3		clwlkfclwwlk.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
4		clwlkfclwwlk.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
5	1, 2, 3, 4	clwlkfclwwlk 26371	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶((𝑉 ClWWalksN 𝐸)‘𝑁))
6		simprl 790	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 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 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 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 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 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 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → (𝐹‘𝑤) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩))
30	16, 29	eqeq12d 2625	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 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	clwlkfclwwlk1hashn 26368	. . . . . . . . 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 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → 𝑁 = (#‘(1st ‘𝑤)))
35		prmnn 15226	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
36	35	3ad2ant3 1077	. . . . . . . . 9 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ)
37	36	adantr 480	. . . . . . . 8 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → 𝑁 ∈ ℕ)
38	17	adantl 481	. . . . . . . 8 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → 𝑤 ∈ 𝐶)
39	1, 2, 3, 4	clwlkf1clwwlklem 26376	. . . . . . . 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)))
40	37, 6, 38, 39	syl3anc 1318	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 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	40	imp 444	. . . . . 6 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 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))
42		simpll1 1093	. . . . . . 7 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → 𝑉 USGrph 𝐸)
43		elrabi 3328	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁} → 𝑢 ∈ (𝑉 ClWalks 𝐸))
44		clwlkswlks 26286	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (𝑉 ClWalks 𝐸) → 𝑢 ∈ (𝑉 Walks 𝐸))
45	43, 44	syl 17	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁} → 𝑢 ∈ (𝑉 Walks 𝐸))
46	45, 3	eleq2s 2706	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐶 → 𝑢 ∈ (𝑉 Walks 𝐸))
47		elrabi 3328	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁} → 𝑤 ∈ (𝑉 ClWalks 𝐸))
48		clwlkswlks 26286	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (𝑉 ClWalks 𝐸) → 𝑤 ∈ (𝑉 Walks 𝐸))
49	47, 48	syl 17	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁} → 𝑤 ∈ (𝑉 Walks 𝐸))
50	49, 3	eleq2s 2706	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐶 → 𝑤 ∈ (𝑉 Walks 𝐸))
51	46, 50	anim12i 588	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (𝑢 ∈ (𝑉 Walks 𝐸) ∧ 𝑤 ∈ (𝑉 Walks 𝐸)))
52	51	ad2antlr 759	. . . . . . 7 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → (𝑢 ∈ (𝑉 Walks 𝐸) ∧ 𝑤 ∈ (𝑉 Walks 𝐸)))
53	1, 2, 3, 4	clwlkfclwwlk1hashn 26368	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝐶 → (#‘(1st ‘𝑢)) = 𝑁)
54	53	eqcomd 2616	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐶 → 𝑁 = (#‘(1st ‘𝑢)))
55	54	adantr 480	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → 𝑁 = (#‘(1st ‘𝑢)))
56	55	ad2antlr 759	. . . . . . 7 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → 𝑁 = (#‘(1st ‘𝑢)))
57		usg2wlkeq 26236	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝑢 ∈ (𝑉 Walks 𝐸) ∧ 𝑤 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝑢))) → (𝑢 = 𝑤 ↔ (𝑁 = (#‘(1st ‘𝑤)) ∧ ∀𝑖 ∈ (0...𝑁)((2nd ‘𝑢)‘𝑖) = ((2nd ‘𝑤)‘𝑖))))
58	42, 52, 56, 57	syl3anc 1318	. . . . . 6 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 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))))
59	34, 41, 58	mpbir2and 959	. . . . 5 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) ∧ ((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩)) → 𝑢 = 𝑤)
60	59	ex 449	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → (((2nd ‘𝑢) substr ⟨0, (#‘(1st ‘𝑢))⟩) = ((2nd ‘𝑤) substr ⟨0, (#‘(1st ‘𝑤))⟩) → 𝑢 = 𝑤))
61	30, 60	sylbid 229	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ (𝑢 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶)) → ((𝐹‘𝑢) = (𝐹‘𝑤) → 𝑢 = 𝑤))
62	61	ralrimivva 2954	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → ∀𝑢 ∈ 𝐶 ∀𝑤 ∈ 𝐶 ((𝐹‘𝑢) = (𝐹‘𝑤) → 𝑢 = 𝑤))
63		dff13 6416	. 2 ⊢ (𝐹:𝐶–1-1→((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝐹:𝐶⟶((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∀𝑢 ∈ 𝐶 ∀𝑤 ∈ 𝐶 ((𝐹‘𝑢) = (𝐹‘𝑤) → 𝑢 = 𝑤)))
64	5, 62, 63	sylanbrc 695	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶–1-1→((𝑉 ClWWalksN 𝐸)‘𝑁))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  Fincfn 7841  0cc0 9815  ℕcn 10897  ...cfz 12197  #chash 12979   substr csubstr 13150  ℙcprime 15223   USGrph cusg 25859   Walks cwalk 26026   ClWalks cclwlk 26275   ClWWalksN cclwwlkn 26277

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-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-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-usgra 25862  df-wlk 26036  df-clwlk 26278  df-clwwlk 26279  df-clwwlkn 26280

This theorem is referenced by:  clwlkf1oclwwlk  26378