Theorem eleclclwwlkn 30146

Description: A member of an equivalence class according to ∼. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 1-May-2021.)

Hypotheses

Ref	Expression
erclwwlkn.w	⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
erclwwlkn.r	⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}

Assertion

Ref	Expression
eleclclwwlkn	⊢ ((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))

Distinct variable groups:   𝑡,𝑊,𝑢   𝑛,𝑁,𝑢,𝑡   𝑛,𝑊   𝑛,𝐺   𝑛,𝑋   𝑛,𝑌

Allowed substitution hints:   𝐵(𝑢,𝑡,𝑛)   ∼ (𝑢,𝑡,𝑛)   𝐺(𝑢,𝑡)   𝑋(𝑢,𝑡)   𝑌(𝑢,𝑡)

Proof of Theorem eleclclwwlkn

Dummy variables 𝑥 𝑦 𝑚 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		erclwwlkn.w	. . . . 5 ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
2		erclwwlkn.r	. . . . 5 ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
3	1, 2	eclclwwlkn1 30145	. . . 4 ⊢ (𝐵 ∈ (𝑊 / ∼ ) → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4		eqeq1 2740	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑌 = (𝑥 cyclShift 𝑛)))
5	4	rexbidv 3161	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛)))
6	5	elrab 3634	. . . . . . . 8 ⊢ (𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛)))
7		oveq2 7375	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑘))
8	7	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑌 = (𝑥 cyclShift 𝑛) ↔ 𝑌 = (𝑥 cyclShift 𝑘)))
9	8	cbvrexvw 3216	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘))
10		eqeq1 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑋 = (𝑥 cyclShift 𝑛)))
11	10	rexbidv 3161	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑋 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛)))
12	11	elrab 3634	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑋 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛)))
13		oveq2 7375	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
14	13	eqeq2d 2747	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑋 = (𝑥 cyclShift 𝑛) ↔ 𝑋 = (𝑥 cyclShift 𝑚)))
15	14	cbvrexvw 3216	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑚))
16	1	eleclclwwlknlem2 30131	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑚)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
17	16	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑚)) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
18	17	rexlimiva 3130	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑚 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑚) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
19	15, 18	sylbi 217	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
20	19	expd 415	. . . . . . . . . . . . . . 15 ⊢ (∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛) → (𝑋 ∈ 𝑊 → (𝑥 ∈ 𝑊 → (∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))))
21	20	impcom 407	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛)) → (𝑥 ∈ 𝑊 → (∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
22	12, 21	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (𝑥 ∈ 𝑊 → (∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
23	22	com12 32	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 → (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
24	23	ad2antlr 728	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑥 ∈ 𝑊) ∧ 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) → (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
25	24	imp 406	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑥 ∈ 𝑊) ∧ 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) ∧ 𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) → (∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
26	9, 25	bitrid 283	. . . . . . . . 9 ⊢ ((((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑥 ∈ 𝑊) ∧ 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) ∧ 𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) → (∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
27	26	anbi2d 631	. . . . . . . 8 ⊢ ((((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑥 ∈ 𝑊) ∧ 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) ∧ 𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) → ((𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛)) ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
28	6, 27	bitrid 283	. . . . . . 7 ⊢ ((((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑥 ∈ 𝑊) ∧ 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) ∧ 𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) → (𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
29	28	ex 412	. . . . . 6 ⊢ (((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑥 ∈ 𝑊) ∧ 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) → (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))))
30		eleq2 2825	. . . . . . . 8 ⊢ (𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
31		eleq2 2825	. . . . . . . . 9 ⊢ (𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
32	31	bibi1d 343	. . . . . . . 8 ⊢ (𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))) ↔ (𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))))
33	30, 32	imbi12d 344	. . . . . . 7 ⊢ (𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))) ↔ (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))))
34	33	adantl 481	. . . . . 6 ⊢ (((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑥 ∈ 𝑊) ∧ 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) → ((𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))) ↔ (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))))
35	29, 34	mpbird 257	. . . . 5 ⊢ (((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑥 ∈ 𝑊) ∧ 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))))
36	35	rexlimdva2 3140	. . . 4 ⊢ (𝐵 ∈ (𝑊 / ∼ ) → (∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))))
37	3, 36	sylbid 240	. . 3 ⊢ (𝐵 ∈ (𝑊 / ∼ ) → (𝐵 ∈ (𝑊 / ∼ ) → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))))
38	37	pm2.43i 52	. 2 ⊢ (𝐵 ∈ (𝑊 / ∼ ) → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))))
39	38	imp 406	1 ⊢ ((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  {crab 3389  {copab 5147  (class class class)co 7367   / cqs 8642  0cc0 11038  ...cfz 13461   cyclShift ccsh 14750   ClWWalksN cclwwlkn 30094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-ec 8645  df-qs 8649  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-fl 13751  df-mod 13829  df-hash 14293  df-word 14476  df-concat 14533  df-substr 14604  df-pfx 14634  df-csh 14751  df-clwwlk 30052  df-clwwlkn 30095

This theorem is referenced by: (None)