Theorem eleclclwwlkn 30005

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 30004	. . . 4 ⊢ (𝐵 ∈ (𝑊 / ∼ ) → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4		eqeq1 2733	. . . . . . . . . 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 3659	. . . . . . . 8 ⊢ (𝑌 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑛)))
7		oveq2 7395	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑘))
8	7	eqeq2d 2740	. . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . 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 3659	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ (𝑋 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑋 = (𝑥 cyclShift 𝑛)))
13		oveq2 7395	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
14	13	eqeq2d 2740	. . . . . . . . . . . . . . . . . 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 29990	. . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . 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 2817	. . . . . . . . 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 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3405  {copab 5169  (class class class)co 7387   / cqs 8670  0cc0 11068  ...cfz 13468   cyclShift ccsh 14753   ClWWalksN cclwwlkn 29953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-ec 8673  df-qs 8677  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-fz 13469  df-fzo 13616  df-fl 13754  df-mod 13832  df-hash 14296  df-word 14479  df-concat 14536  df-substr 14606  df-pfx 14636  df-csh 14754  df-clwwlk 29911  df-clwwlkn 29954

This theorem is referenced by: (None)