Theorem eleclclwwlkn 30077

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 30076	. . . 4 ⊢ (𝐵 ∈ (𝑊 / ∼ ) → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4		eqeq1 2737	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑌 = (𝑥 cyclShift 𝑛)))
5	4	rexbidv 3157	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛)))
6	5	elrab 3643	. . . . . . . 8 ⊢ (𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛)))
7		oveq2 7363	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑘))
8	7	eqeq2d 2744	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑌 = (𝑥 cyclShift 𝑛) ↔ 𝑌 = (𝑥 cyclShift 𝑘)))
9	8	cbvrexvw 3212	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑘))
10		eqeq1 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑋 = (𝑥 cyclShift 𝑛)))
11	10	rexbidv 3157	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑋 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛)))
12	11	elrab 3643	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑋 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛)))
13		oveq2 7363	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
14	13	eqeq2d 2744	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑋 = (𝑥 cyclShift 𝑛) ↔ 𝑋 = (𝑥 cyclShift 𝑚)))
15	14	cbvrexvw 3212	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑚))
16	1	eleclclwwlknlem2 30062	. . . . . . . . . . . . . . . . . . 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 3126	. . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . 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 630	. . . . . . . 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 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 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 3136	. . . 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 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3057  {crab 3396  {copab 5157  (class class class)co 7355   / cqs 8630  0cc0 11017  ...cfz 13414   cyclShift ccsh 14702   ClWWalksN cclwwlkn 30025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094  ax-pre-sup 11095

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-ec 8633  df-qs 8637  df-map 8761  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9337  df-inf 9338  df-card 9843  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-div 11786  df-nn 12137  df-2 12199  df-n0 12393  df-z 12480  df-uz 12743  df-rp 12897  df-fz 13415  df-fzo 13562  df-fl 13703  df-mod 13781  df-hash 14245  df-word 14428  df-concat 14485  df-substr 14556  df-pfx 14586  df-csh 14703  df-clwwlk 29983  df-clwwlkn 30026

This theorem is referenced by: (None)