Theorem umgrhashecclwwlk 41262

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 equals this length (in an undirected simple graph). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 1-May-2021.)

Hypotheses

Ref	Expression
erclwwlksn.w	⊢ 𝑊 = (𝑁 ClWWalkSN 𝐺)
erclwwlksn.r	⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}

Assertion

Ref	Expression
umgrhashecclwwlk	⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ∼ ) → (#‘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 umgrhashecclwwlk

Dummy variables 𝑥 𝑦 𝑚 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		erclwwlksn.w	. . . . 5 ⊢ 𝑊 = (𝑁 ClWWalkSN 𝐺)
2		erclwwlksn.r	. . . . 5 ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
3	1, 2	eclclwwlksn1 41259	. . . 4 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4		rabeq 3166	. . . . . . . . . 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 ⊢ (((𝐺 ∈ UMGraph ∧ 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 15226	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
7	6	nnnn0d 11228	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ0)
8	7	adantl 481	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ0)
9	1	eleq2i 2680	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ (𝑁 ClWWalkSN 𝐺))
10	9	biimpi 205	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → 𝑥 ∈ (𝑁 ClWWalkSN 𝐺))
11		clwwlksnscsh 41247	. . . . . . . . . 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)})
12	8, 10, 11	syl2an 493	. . . . . . . . 9 ⊢ (((𝐺 ∈ UMGraph ∧ 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)})
13	5, 12	eqtrd 2644	. . . . . . . 8 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥 ∈ 𝑊) → {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
14	13	eqeq2d 2620	. . . . . . 7 ⊢ (((𝐺 ∈ UMGraph ∧ 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)}))
15	6	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ)
16		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
17	16	clwwlknbp 41193	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁))
18		simpll 786	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ∈ Word (Vtx‘𝐺))
19		elnnne0 11183	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
20		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 = (#‘𝑥) → (𝑁 = 0 ↔ (#‘𝑥) = 0))
21	20	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑥) = 𝑁 → (𝑁 = 0 ↔ (#‘𝑥) = 0))
22		hasheq0 13015	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
23	21, 22	sylan9bbr 733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑁 = 0 ↔ 𝑥 = ∅))
24	23	necon3bid 2826	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑁 ≠ 0 ↔ 𝑥 ≠ ∅))
25	24	biimpcd 238	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ≠ 0 → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
26	19, 25	simplbiim 657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
27	26	impcom 445	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ≠ ∅)
28		simplr 788	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (#‘𝑥) = 𝑁)
29	28	eqcomd 2616	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑁 = (#‘𝑥))
30	18, 27, 29	3jca 1235	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)))
31	30	ex 449	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑁 ∈ ℕ → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
32	17, 31	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → (𝑁 ∈ ℕ → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
33	32	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
34	9, 33	syl5bi 231	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
35	15, 34	syl 17	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
36	35	imp 444	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)))
37		scshwfzeqfzo 13423	. . . . . . . . . 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)})
38	36, 37	syl 17	. . . . . . . . 9 ⊢ (((𝐺 ∈ UMGraph ∧ 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)})
39	38	eqeq2d 2620	. . . . . . . 8 ⊢ (((𝐺 ∈ UMGraph ∧ 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)}))
40		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}))
41		simprl 790	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ)) → 𝐺 ∈ UMGraph )
42		prmuz2 15246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑥) ∈ ℙ → (#‘𝑥) ∈ (ℤ≥‘2))
43	42	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ) → (#‘𝑥) ∈ (ℤ≥‘2))
44	43	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ)) → (#‘𝑥) ∈ (ℤ≥‘2))
45		simplr 788	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ)) → 𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺))
46		umgr2cwwkdifex 41249	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ (ℤ≥‘2) ∧ 𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺)) → ∃𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) ≠ (𝑥‘0))
47	41, 44, 45, 46	syl3anc 1318	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ)) → ∃𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) ≠ (𝑥‘0))
48		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
49	48	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
50	49	cbvrexv 3148	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
51		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ 𝑢 = (𝑥 cyclShift 𝑚)))
52		eqcom 2617	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢)
53	51, 52	syl6bb 275	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢))
54	53	rexbidv 3034	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑢 → (∃𝑚 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0..^(#‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
55	50, 54	syl5bb 271	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑢 → (∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(#‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
56	55	cbvrabv 3172	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} = {𝑢 ∈ Word (Vtx‘𝐺) ∣ ∃𝑚 ∈ (0..^(#‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢}
57	56	cshwshashnsame 15648	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) ∈ ℙ) → (∃𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) ≠ (𝑥‘0) → (#‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥)))
58	57	ad2ant2rl 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ)) → (∃𝑖 ∈ (0..^(#‘𝑥))(𝑥‘𝑖) ≠ (𝑥‘0) → (#‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥)))
59	47, 58	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ)) → (#‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥))
60	40, 59	sylan9eqr 2666	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ)) ∧ 𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) → (#‘𝑈) = (#‘𝑥))
61	60	exp41 636	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → (𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺) → ((𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥)))))
62	61	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺) → ((𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥)))))
63		oveq1 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (#‘𝑥) → (𝑁 ClWWalkSN 𝐺) = ((#‘𝑥) ClWWalkSN 𝐺))
64	63	eleq2d 2673	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (#‘𝑥) → (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) ↔ 𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺)))
65		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (#‘𝑥) → (𝑁 ∈ ℙ ↔ (#‘𝑥) ∈ ℙ))
66	65	anbi2d 736	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (#‘𝑥) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ↔ (𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ)))
67		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 = (#‘𝑥) → (0..^𝑁) = (0..^(#‘𝑥)))
68	67	rexeqdv 3122	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (#‘𝑥) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
69	68	rabbidv 3164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (#‘𝑥) → {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)})
70	69	eqeq2d 2620	. . . . . . . . . . . . . . . . 17 ⊢ (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)}))
71		eqeq2 2621	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (#‘𝑥) → ((#‘𝑈) = 𝑁 ↔ (#‘𝑈) = (#‘𝑥)))
72	70, 71	imbi12d 333	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (#‘𝑥) → ((𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁) ↔ (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥))))
73	66, 72	imbi12d 333	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (#‘𝑥) → (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁)) ↔ ((𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥)))))
74	64, 73	imbi12d 333	. . . . . . . . . . . . . 14 ⊢ (𝑁 = (#‘𝑥) → ((𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))) ↔ (𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺) → ((𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥))))))
75	74	eqcoms 2618	. . . . . . . . . . . . 13 ⊢ ((#‘𝑥) = 𝑁 → ((𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))) ↔ (𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺) → ((𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥))))))
76	75	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → ((𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))) ↔ (𝑥 ∈ ((#‘𝑥) ClWWalkSN 𝐺) → ((𝐺 ∈ UMGraph ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥))))))
77	62, 76	mpbird 246	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))))
78	17, 77	mpcom 37	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁)))
79	78, 1	eleq2s 2706	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁)))
80	79	impcom 445	. . . . . . . 8 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))
81	39, 80	sylbid 229	. . . . . . 7 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))
82	14, 81	sylbid 229	. . . . . 6 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))
83	82	rexlimdva 3013	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))
84	83	com12 32	. . . 4 ⊢ (∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (#‘𝑈) = 𝑁))
85	3, 84	syl6bi 242	. . 3 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (#‘𝑈) = 𝑁)))
86	85	pm2.43i 50	. 2 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (#‘𝑈) = 𝑁))
87	86	com12 32	1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ∼ ) → (#‘𝑈) = 𝑁))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  ∅c0 3874  {copab 4642  ‘cfv 5804  (class class class)co 6549   / cqs 7628  0cc0 9815  ℕcn 10897  2c2 10947  ℕ₀cn0 11169  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   cyclShift ccsh 13385  ℙcprime 15223  Vtxcvtx 25673   UMGraph cumgr 25748   ClWWalkSN cclwwlksn 41184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-substr 13158  df-reps 13161  df-csh 13386  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-gcd 15055  df-prm 15224  df-phi 15309  df-umgr 25750  df-edga 25793  df-clwwlks 41185  df-clwwlksn 41186

This theorem is referenced by:  fusgrhashclwwlkn  41263