Theorem umgrhashecclwwlk 30014

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
erclwwlkn.w	⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
erclwwlkn.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		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 30011	. . . 4 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4		rabeq 3423	. . . . . . . . . 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 16651	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
7	6	nnnn0d 12510	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ0)
8	7	adantl 481	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ0)
9	1	eleq2i 2821	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ (𝑁 ClWWalksN 𝐺))
10	9	biimpi 216	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → 𝑥 ∈ (𝑁 ClWWalksN 𝐺))
11		clwwlknscsh 29998	. . . . . . . . . 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 596	. . . . . . . . 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 2765	. . . . . . . 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 2741	. . . . . . 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		simpll 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ∈ Word (Vtx‘𝐺))
17		elnnne0 12463	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
18		eqeq1 2734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 = (♯‘𝑥) → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
19	18	eqcoms 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑥) = 𝑁 → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
20		hasheq0 14335	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
21	19, 20	sylan9bbr 510	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 = 0 ↔ 𝑥 = ∅))
22	21	necon3bid 2970	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ≠ 0 ↔ 𝑥 ≠ ∅))
23	22	biimpcd 249	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ≠ 0 → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
24	17, 23	simplbiim 504	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
25	24	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ≠ ∅)
26		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (♯‘𝑥) = 𝑁)
27	26	eqcomd 2736	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑁 = (♯‘𝑥))
28	16, 25, 27	3jca 1128	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)))
29	28	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ∈ ℕ → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
30		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
31	30	clwwlknbp 29971	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
32	29, 31	syl11 33	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
33	9, 32	biimtrid 242	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
34	15, 33	syl 17	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
35	34	imp 406	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)))
36		scshwfzeqfzo 14799	. . . . . . . . . 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)})
37	35, 36	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)})
38	37	eqeq2d 2741	. . . . . . . 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)}))
39		fveq2 6861	. . . . . . . . . . . . . . 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)}))
40		simprl 770	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)) → 𝐺 ∈ UMGraph)
41		prmuz2 16673	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑥) ∈ ℙ → (♯‘𝑥) ∈ (ℤ≥‘2))
42	41	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ) → (♯‘𝑥) ∈ (ℤ≥‘2))
43	42	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)) → (♯‘𝑥) ∈ (ℤ≥‘2))
44		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)) → 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺))
45		umgr2cwwkdifex 30001	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ (ℤ≥‘2) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) → ∃𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) ≠ (𝑥‘0))
46	40, 43, 44, 45	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)) → ∃𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) ≠ (𝑥‘0))
47		oveq2 7398	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
48	47	eqeq2d 2741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
49	48	cbvrexvw 3217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
50		eqeq1 2734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ 𝑢 = (𝑥 cyclShift 𝑚)))
51		eqcom 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢)
52	50, 51	bitrdi 287	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢))
53	52	rexbidv 3158	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑢 → (∃𝑚 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
54	49, 53	bitrid 283	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑢 → (∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
55	54	cbvrabv 3419	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} = {𝑢 ∈ Word (Vtx‘𝐺) ∣ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢}
56	55	cshwshashnsame 17081	. . . . . . . . . . . . . . . . 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)))
57	56	ad2ant2rl 749	. . . . . . . . . . . . . . . 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)))
58	46, 57	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))
59	39, 58	sylan9eqr 2787	. . . . . . . . . . . . . 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))
60	59	exp41 434	. . . . . . . . . . . . 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)))))
61	60	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)))))
62		oveq1 7397	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → (𝑁 ClWWalksN 𝐺) = ((♯‘𝑥) ClWWalksN 𝐺))
63	62	eleq2d 2815	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (♯‘𝑥) → (𝑥 ∈ (𝑁 ClWWalksN 𝐺) ↔ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)))
64		eleq1 2817	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘𝑥) → (𝑁 ∈ ℙ ↔ (♯‘𝑥) ∈ ℙ))
65	64	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ↔ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)))
66		oveq2 7398	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 = (♯‘𝑥) → (0..^𝑁) = (0..^(♯‘𝑥)))
67	66	rexeqdv 3302	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑥) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
68	67	rabbidv 3416	. . . . . . . . . . . . . . . . . 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)})
69	68	eqeq2d 2741	. . . . . . . . . . . . . . . . 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)}))
70		eqeq2 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘𝑥) → ((♯‘𝑈) = 𝑁 ↔ (♯‘𝑈) = (♯‘𝑥)))
71	69, 70	imbi12d 344	. . . . . . . . . . . . . . . 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))))
72	65, 71	imbi12d 344	. . . . . . . . . . . . . . 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)))))
73	63, 72	imbi12d 344	. . . . . . . . . . . . . 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))))))
74	73	eqcoms 2738	. . . . . . . . . . . . 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))))))
75	74	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))))))
76	61, 75	mpbird 257	. . . . . . . . . . 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))))
77	31, 76	mpcom 38	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁)))
78	77, 1	eleq2s 2847	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁)))
79	78	impcom 407	. . . . . . . 8 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁))
80	38, 79	sylbid 240	. . . . . . 7 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁))
81	14, 80	sylbid 240	. . . . . 6 ⊢ (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁))
82	81	rexlimdva 3135	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁))
83	82	com12 32	. . . 4 ⊢ (∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (♯‘𝑈) = 𝑁))
84	3, 83	biimtrdi 253	. . 3 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (♯‘𝑈) = 𝑁)))
85	84	pm2.43i 52	. 2 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (♯‘𝑈) = 𝑁))
86	85	com12 32	1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ∼ ) → (♯‘𝑈) = 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054  {crab 3408  ∅c0 4299  {copab 5172  ‘cfv 6514  (class class class)co 7390   / cqs 8673  0cc0 11075  ℕcn 12193  2c2 12248  ℕ₀cn0 12449  ℤ_≥cuz 12800  ...cfz 13475  ..^cfzo 13622  ♯chash 14302  Word cword 14485   cyclShift ccsh 14760  ℙcprime 16648  Vtxcvtx 28930  UMGraphcumgr 29015   ClWWalksN cclwwlkn 29960

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-disj 5078  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8674  df-ec 8676  df-qs 8680  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-rp 12959  df-ico 13319  df-fz 13476  df-fzo 13623  df-fl 13761  df-mod 13839  df-seq 13974  df-exp 14034  df-hash 14303  df-word 14486  df-lsw 14535  df-concat 14543  df-substr 14613  df-pfx 14643  df-reps 14741  df-csh 14761  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-sum 15660  df-dvds 16230  df-gcd 16472  df-prm 16649  df-phi 16743  df-edg 28982  df-umgr 29017  df-clwwlk 29918  df-clwwlkn 29961

This theorem is referenced by:  fusgrhashclwwlkn  30015