Theorem eleclclwwlkn 30546

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 30545	. . . 4 ⊢ (𝐵 ∈ (𝑊 / ∼ ) → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑌 = (𝑥 cyclShift 𝑛)))
5	4	rexbidv 3186	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛)))
6	5	elrab 3645	. . . . . . . 8 ⊢ (𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛)))
7		oveq2 7421	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑘))
8	7	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑌 = (𝑥 cyclShift 𝑛) ↔ 𝑌 = (𝑥 cyclShift 𝑘)))
9	8	cbvrexvw 3241	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘))
10		eqeq1 2764	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑋 = (𝑥 cyclShift 𝑛)))
11	10	rexbidv 3186	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑋 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛)))
12	11	elrab 3645	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑋 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛)))
13		oveq2 7421	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
14	13	eqeq2d 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑋 = (𝑥 cyclShift 𝑛) ↔ 𝑋 = (𝑥 cyclShift 𝑚)))
15	14	cbvrexvw 3241	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑚))
16	1	eleclclwwlknlem2 30531	. . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . 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 421	. . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . 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 220	. . . . . . . . . . . . 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 33	. . . . . . . . . . . 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 740	. . . . . . . . . . 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 412	. . . . . . . . . 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 286	. . . . . . . . 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 642	. . . . . . . 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 286	. . . . . . 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 418	. . . . . 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 2849	. . . . . . . 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 2849	. . . . . . . . 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 346	. . . . . . . 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 347	. . . . . . 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 487	. . . . . 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 260	. . . . 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 3165	. . . 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 243	. . 3 ⊢ (𝐵 ∈ (𝑊 / ∼ ) → (𝐵 ∈ (𝑊 / ∼ ) → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))))
38	37	pm2.43i 53	. 2 ⊢ (𝐵 ∈ (𝑊 / ∼ ) → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))))
39	38	imp 412	1 ⊢ ((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  {crab 3412  {copab 5167  (class class class)co 7413   / cqs 8695  0cc0 11124  ...cfz 13561   cyclShift ccsh 14859   ClWWalksN cclwwlkn 30494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-ec 8698  df-qs 8702  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-n0 12529  df-z 12616  df-uz 12888  df-rp 13043  df-fz 13562  df-fzo 13710  df-fl 13853  df-mod 13931  df-hash 14395  df-word 14579  df-concat 14636  df-substr 14709  df-pfx 14741  df-csh 14860  df-clwwlk 30452  df-clwwlkn 30495

This theorem is used by: (None)