Theorem hashecclwwlkn1 30333

Description: The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number is 1 or equals this length. (Contributed by Alexander van der Vekens, 17-Jun-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
hashecclwwlkn1	⊢ ((𝑁 ∈ ℙ ∧ 𝑈 ∈ (𝑊 / ∼ )) → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))

Distinct variable groups:   𝑡,𝑊,𝑢   𝑛,𝑁,𝑢,𝑡   𝑛,𝑊   𝑛,𝐺,𝑢   𝑈,𝑛,𝑢

Allowed substitution hints:   ∼ (𝑢,𝑡,𝑛)   𝑈(𝑡)   𝐺(𝑡)

Proof of Theorem hashecclwwlkn1

Dummy variables 𝑥 𝑦 𝑚 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 30331	. . . 4 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4		rabeq 3431	. . . . . . . . . 10 ⊢ (𝑊 = (𝑁 ClWWalksN 𝐺) → {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
5	1, 4	mp1i 14	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
6		prmnn 16720	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
7	6	nnnn0d 12553	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ0)
8	1	eleq2i 2857	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ (𝑁 ClWWalksN 𝐺))
9	8	biimpi 219	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → 𝑥 ∈ (𝑁 ClWWalksN 𝐺))
10		clwwlknscsh 30318	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
11	7, 9, 10	syl2an 607	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
12	5, 11	eqtrd 2800	. . . . . . . 8 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
13	12	eqeq2d 2776	. . . . . . 7 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
14		simpll 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ∈ Word (Vtx‘𝐺))
15		elnnne0 12506	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
16		eqeq1 2769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 = (♯‘𝑥) → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
17	16	eqcoms 2773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑥) = 𝑁 → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
18		hasheq0 14387	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
19	17, 18	sylan9bbr 519	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 = 0 ↔ 𝑥 = ∅))
20	19	necon3bid 3004	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ≠ 0 ↔ 𝑥 ≠ ∅))
21	20	biimpcd 252	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ≠ 0 → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
22	15, 21	simplbiim 513	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
23	22	impcom 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ≠ ∅)
24		simplr 780	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (♯‘𝑥) = 𝑁)
25	24	eqcomd 2771	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑁 = (♯‘𝑥))
26	14, 23, 25	3jca 1144	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)))
27	26	ex 417	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ∈ ℕ → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
28		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
29	28	clwwlknbp 30291	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
30	27, 29	syl11 34	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
31	8, 30	biimtrid 245	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
32	6, 31	syl 18	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
33	32	imp 411	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)))
34		scshwfzeqfzo 14851	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)) → {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
35	33, 34	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
36	35	eqeq2d 2776	. . . . . . . 8 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
37		oveq2 7408	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
38	37	eqeq2d 2776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
39	38	cbvrexvw 3244	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
40		eqeq1 2769	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ 𝑢 = (𝑥 cyclShift 𝑚)))
41		eqcom 2772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢)
42	40, 41	bitrdi 290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢))
43	42	rexbidv 3189	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑢 → (∃𝑚 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
44	39, 43	bitrid 286	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑢 → (∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
45	44	cbvrabv 3427	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} = {𝑢 ∈ Word (Vtx‘𝐺) ∣ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢}
46	45	cshwshash 17152	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) → ((♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥) ∨ (♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1))
47	46	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) → ((♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥) ∨ (♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1))
48	47	orcomd 884	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) → ((♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1 ∨ (♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥)))
49		fveqeq2 6880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ↔ (♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1))
50		fveqeq2 6880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = (♯‘𝑥) ↔ (♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥)))
51	49, 50	orbi12d 931	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥)) ↔ ((♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1 ∨ (♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥))))
52	51	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) → (((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥)) ↔ ((♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1 ∨ (♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥))))
53	48, 52	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥)))
54	53	ex 417	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥))))
55	54	ex 417	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → ((♯‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥)))))
56	55	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → ((♯‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥)))))
57		eleq1 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (♯‘𝑥) → (𝑁 ∈ ℙ ↔ (♯‘𝑥) ∈ ℙ))
58		oveq2 7408	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑥) → (0..^𝑁) = (0..^(♯‘𝑥)))
59	58	rexeqdv 3324	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘𝑥) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
60	59	rabbidv 3424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘𝑥) → {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)})
61	60	eqeq2d 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}))
62		eqeq2 2777	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘𝑥) → ((♯‘𝑈) = 𝑁 ↔ (♯‘𝑈) = (♯‘𝑥)))
63	62	orbi2d 928	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → (((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁) ↔ ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥))))
64	61, 63	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (♯‘𝑥) → ((𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)) ↔ (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥)))))
65	57, 64	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑁 = (♯‘𝑥) → ((𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))) ↔ ((♯‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥))))))
66	65	eqcoms 2773	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 𝑁 → ((𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))) ↔ ((♯‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥))))))
67	66	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → ((𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))) ↔ ((♯‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥))))))
68	56, 67	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))))
69	29, 68	syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))))
70	69, 1	eleq2s 2883	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → (𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))))
71	70	impcom 412	. . . . . . . 8 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)))
72	36, 71	sylbid 243	. . . . . . 7 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)))
73	13, 72	sylbid 243	. . . . . 6 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)))
74	73	rexlimdva 3166	. . . . 5 ⊢ (𝑁 ∈ ℙ → (∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)))
75	74	com12 33	. . . 4 ⊢ (∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (𝑁 ∈ ℙ → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)))
76	3, 75	biimtrdi 256	. . 3 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) → (𝑁 ∈ ℙ → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))))
77	76	pm2.43i 53	. 2 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑁 ∈ ℙ → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)))
78	77	impcom 412	1 ⊢ ((𝑁 ∈ ℙ ∧ 𝑈 ∈ (𝑊 / ∼ )) → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  {crab 3417  ∅c0 4288  {copab 5166  ‘cfv 6525  (class class class)co 7400   / cqs 8681  0cc0 11088  1c1 11089  ℕcn 12221  ℕ₀cn0 12492  ...cfz 13523  ..^cfzo 13670  ♯chash 14354  Word cword 14538   cyclShift ccsh 14813  ℙcprime 16717  Vtxcvtx 29251   ClWWalksN cclwwlkn 30280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-disj 5072  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-er 8682  df-ec 8684  df-qs 8688  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-inf 9391  df-oi 9460  df-dju 9875  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12222  df-2 12291  df-3 12292  df-n0 12493  df-xnn0 12566  df-z 12580  df-uz 12851  df-rp 13005  df-ico 13366  df-fz 13524  df-fzo 13671  df-fl 13813  df-mod 13891  df-seq 14026  df-exp 14086  df-hash 14355  df-word 14539  df-lsw 14588  df-concat 14596  df-substr 14667  df-pfx 14697  df-reps 14794  df-csh 14814  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15527  df-sum 15726  df-dvds 16299  df-gcd 16541  df-prm 16718  df-phi 16813  df-clwwlk 30238  df-clwwlkn 30281

This theorem is referenced by: (None)