Theorem eleclclwwlkn 29367

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 29366	. . . 4 ⊢ (𝐵 ∈ (𝑊 / ∼ ) → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4		eqeq1 2736	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑌 = (𝑥 cyclShift 𝑛)))
5	4	rexbidv 3178	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛)))
6	5	elrab 3683	. . . . . . . 8 ⊢ (𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛)))
7		oveq2 7419	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑘))
8	7	eqeq2d 2743	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑌 = (𝑥 cyclShift 𝑛) ↔ 𝑌 = (𝑥 cyclShift 𝑘)))
9	8	cbvrexvw 3235	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘))
10		eqeq1 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑋 = (𝑥 cyclShift 𝑛)))
11	10	rexbidv 3178	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑋 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛)))
12	11	elrab 3683	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑋 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛)))
13		oveq2 7419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
14	13	eqeq2d 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑋 = (𝑥 cyclShift 𝑛) ↔ 𝑋 = (𝑥 cyclShift 𝑚)))
15	14	cbvrexvw 3235	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑚))
16	1	eleclclwwlknlem2 29352	. . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . 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 216	. . . . . . . . . . . . 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 725	. . . . . . . . . . 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 407	. . . . . . . . . 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 282	. . . . . . . . 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 629	. . . . . . . 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 282	. . . . . . 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 413	. . . . . 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 2822	. . . . . . . 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 2822	. . . . . . . . 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 482	. . . . . 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 256	. . . . 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 3157	. . . 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 239	. . 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 407	1 ⊢ ((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  {crab 3432  {copab 5210  (class class class)co 7411   / cqs 8704  0cc0 11112  ...cfz 13486   cyclShift ccsh 14740   ClWWalksN cclwwlkn 29315

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-ec 8707  df-qs 8711  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-n0 12475  df-z 12561  df-uz 12825  df-rp 12977  df-fz 13487  df-fzo 13630  df-fl 13759  df-mod 13837  df-hash 14293  df-word 14467  df-concat 14523  df-substr 14593  df-pfx 14623  df-csh 14741  df-clwwlk 29273  df-clwwlkn 29316

This theorem is referenced by: (None)