Theorem umgrhashecclwwlk 30107

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 30104	. . . 4 ⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4		rabeq 3448	. . . . . . . . . 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 16708	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
7	6	nnnn0d 12585	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ0)
8	7	adantl 481	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ0)
9	1	eleq2i 2831	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ (𝑁 ClWWalksN 𝐺))
10	9	biimpi 216	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → 𝑥 ∈ (𝑁 ClWWalksN 𝐺))
11		clwwlknscsh 30091	. . . . . . . . . 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 2775	. . . . . . . 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 2746	. . . . . . 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 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ∈ Word (Vtx‘𝐺))
17		elnnne0 12538	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
18		eqeq1 2739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 = (♯‘𝑥) → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
19	18	eqcoms 2743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑥) = 𝑁 → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
20		hasheq0 14399	. . . . . . . . . . . . . . . . . . . . 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 2983	. . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (♯‘𝑥) = 𝑁)
27	26	eqcomd 2741	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑁 = (♯‘𝑥))
28	16, 25, 27	3jca 1127	. . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . 15 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
31	30	clwwlknbp 30064	. . . . . . . . . . . . . 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 14862	. . . . . . . . . 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 2746	. . . . . . . 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 6907	. . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)) → 𝐺 ∈ UMGraph)
41		prmuz2 16730	. . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)) → 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺))
45		umgr2cwwkdifex 30094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ (ℤ≥‘2) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) → ∃𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) ≠ (𝑥‘0))
46	40, 43, 44, 45	syl3anc 1370	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)) → ∃𝑖 ∈ (0..^(♯‘𝑥))(𝑥‘𝑖) ≠ (𝑥‘0))
47		oveq2 7439	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
48	47	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
49	48	cbvrexvw 3236	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
50		eqeq1 2739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ 𝑢 = (𝑥 cyclShift 𝑚)))
51		eqcom 2742	. . . . . . . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . . . . . 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 17138	. . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → (𝑁 ClWWalksN 𝐺) = ((♯‘𝑥) ClWWalksN 𝐺))
63	62	eleq2d 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = (♯‘𝑥) → (𝑥 ∈ (𝑁 ClWWalksN 𝐺) ↔ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)))
64		eleq1 2827	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘𝑥) → (𝑁 ∈ ℙ ↔ (♯‘𝑥) ∈ ℙ))
65	64	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ↔ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)))
66		oveq2 7439	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 = (♯‘𝑥) → (0..^𝑁) = (0..^(♯‘𝑥)))
67	66	rexeqdv 3325	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑥) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
68	67	rabbidv 3441	. . . . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . 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 2857	. . . . . . . . 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 3153	. . . . 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 1537   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068  {crab 3433  ∅c0 4339  {copab 5210  ‘cfv 6563  (class class class)co 7431   / cqs 8743  0cc0 11153  ℕcn 12264  2c2 12319  ℕ₀cn0 12524  ℤ_≥cuz 12876  ...cfz 13544  ..^cfzo 13691  ♯chash 14366  Word cword 14549   cyclShift ccsh 14823  ℙcprime 16705  Vtxcvtx 29028  UMGraphcumgr 29113   ClWWalksN cclwwlkn 30053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-ico 13390  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-hash 14367  df-word 14550  df-lsw 14598  df-concat 14606  df-substr 14676  df-pfx 14706  df-reps 14804  df-csh 14824  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-dvds 16288  df-gcd 16529  df-prm 16706  df-phi 16800  df-edg 29080  df-umgr 29115  df-clwwlk 30011  df-clwwlkn 30054

This theorem is referenced by:  fusgrhashclwwlkn  30108