Theorem hashecclwwlkn1 27492

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 27490	. . . 4 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4		rabeq 3389	. . . . . . . . . 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 13	. . . . . . . . 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 15803	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
7	6	nnnn0d 11707	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ0)
8	1	eleq2i 2851	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ (𝑁 ClWWalksN 𝐺))
9	8	biimpi 208	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → 𝑥 ∈ (𝑁 ClWWalksN 𝐺))
10		clwwlknscsh 27477	. . . . . . . . . 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 589	. . . . . . . . 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 2814	. . . . . . . 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 2788	. . . . . . 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 757	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ∈ Word (Vtx‘𝐺))
15		elnnne0 11663	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
16		eqeq1 2782	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 = (♯‘𝑥) → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
17	16	eqcoms 2786	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑥) = 𝑁 → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
18		hasheq0 13475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
19	17, 18	sylan9bbr 506	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 = 0 ↔ 𝑥 = ∅))
20	19	necon3bid 3013	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ≠ 0 ↔ 𝑥 ≠ ∅))
21	20	biimpcd 241	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ≠ 0 → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
22	15, 21	simplbiim 500	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
23	22	impcom 398	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ≠ ∅)
24		simplr 759	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (♯‘𝑥) = 𝑁)
25	24	eqcomd 2784	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑁 = (♯‘𝑥))
26	14, 23, 25	3jca 1119	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)))
27	26	ex 403	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ∈ ℕ → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
28		eqid 2778	. . . . . . . . . . . . . . 15 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
29	28	clwwlknbp 27441	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
30	27, 29	syl11 33	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
31	8, 30	syl5bi 234	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
32	6, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
33	32	imp 397	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)))
34		scshwfzeqfzo 13983	. . . . . . . . . 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 17	. . . . . . . . 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 2788	. . . . . . . 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 6932	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
38	37	eqeq2d 2788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
39	38	cbvrexv 3368	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
40		eqeq1 2782	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ 𝑢 = (𝑥 cyclShift 𝑚)))
41		eqcom 2785	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢)
42	40, 41	syl6bb 279	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢))
43	42	rexbidv 3237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑢 → (∃𝑚 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
44	39, 43	syl5bb 275	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑢 → (∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
45	44	cbvrabv 3396	. . . . . . . . . . . . . . . . . . 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 16221	. . . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . . 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 860	. . . . . . . . . . . . . . . 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 6457	. . . . . . . . . . . . . . . . . 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 6457	. . . . . . . . . . . . . . . . . 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 905	. . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . . 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 403	. . . . . . . . . . . . . 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 403	. . . . . . . . . . . . 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 474	. . . . . . . . . . . 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 2847	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (♯‘𝑥) → (𝑁 ∈ ℙ ↔ (♯‘𝑥) ∈ ℙ))
58		oveq2 6932	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑥) → (0..^𝑁) = (0..^(♯‘𝑥)))
59	58	rexeqdv 3341	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘𝑥) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
60	59	rabbidv 3386	. . . . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘𝑥) → ((♯‘𝑈) = 𝑁 ↔ (♯‘𝑈) = (♯‘𝑥)))
63	62	orbi2d 902	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → (((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁) ↔ ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = (♯‘𝑥))))
64	61, 63	imbi12d 336	. . . . . . . . . . . . . . 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 336	. . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . 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 475	. . . . . . . . . . . 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 249	. . . . . . . . . . 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 17	. . . . . . . . . 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 2877	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → (𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))))
71	70	impcom 398	. . . . . . . 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 232	. . . . . . 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 232	. . . . . 6 ⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)))
74	73	rexlimdva 3213	. . . . 5 ⊢ (𝑁 ∈ ℙ → (∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)))
75	74	com12 32	. . . 4 ⊢ (∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (𝑁 ∈ ℙ → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)))
76	3, 75	syl6bi 245	. . 3 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) → (𝑁 ∈ ℙ → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))))
77	76	pm2.43i 52	. 2 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑁 ∈ ℙ → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)))
78	77	impcom 398	1 ⊢ ((𝑁 ∈ ℙ ∧ 𝑈 ∈ (𝑊 / ∼ )) → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∃wrex 3091  {crab 3094  ∅c0 4141  {copab 4950  ‘cfv 6137  (class class class)co 6924   / cqs 8027  0cc0 10274  1c1 10275  ℕcn 11379  ℕ₀cn0 11647  ...cfz 12648  ..^cfzo 12789  ♯chash 13441  Word cword 13605   cyclShift ccsh 13940  ℙcprime 15800  Vtxcvtx 26361   ClWWalksN cclwwlkn 27430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-disj 4857  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-ec 8030  df-qs 8034  df-map 8144  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-sup 8638  df-inf 8639  df-oi 8706  df-card 9100  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-n0 11648  df-xnn0 11720  df-z 11734  df-uz 11998  df-rp 12143  df-ico 12498  df-fz 12649  df-fzo 12790  df-fl 12917  df-mod 12993  df-seq 13125  df-exp 13184  df-hash 13442  df-word 13606  df-lsw 13659  df-concat 13667  df-substr 13737  df-pfx 13786  df-reps 13921  df-csh 13942  df-cj 14252  df-re 14253  df-im 14254  df-sqrt 14388  df-abs 14389  df-clim 14636  df-sum 14834  df-dvds 15397  df-gcd 15633  df-prm 15801  df-phi 15886  df-clwwlk 27379  df-clwwlkn 27431

This theorem is referenced by: (None)