Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  umgrhashecclwwlk Structured version   Visualization version   GIF version

Theorem umgrhashecclwwlk 27873
 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⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃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.
StepHypRef Expression
1 erclwwlkn.w . . . . 5 𝑊 = (𝑁 ClWWalksN 𝐺)
2 erclwwlkn.r . . . . 5 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
31, 2eclclwwlkn1 27870 . . . 4 (𝑈 ∈ (𝑊 / ) → (𝑈 ∈ (𝑊 / ) ↔ ∃𝑥𝑊 𝑈 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4 rabeq 3431 . . . . . . . . . 10 (𝑊 = (𝑁 ClWWalksN 𝐺) → {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
51, 4mp1i 13 . . . . . . . . 9 (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥𝑊) → {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
6 prmnn 16011 . . . . . . . . . . . 12 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
76nnnn0d 11946 . . . . . . . . . . 11 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ0)
87adantl 485 . . . . . . . . . 10 ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ0)
91eleq2i 2881 . . . . . . . . . . 11 (𝑥𝑊𝑥 ∈ (𝑁 ClWWalksN 𝐺))
109biimpi 219 . . . . . . . . . 10 (𝑥𝑊𝑥 ∈ (𝑁 ClWWalksN 𝐺))
11 clwwlknscsh 27857 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑥 ∈ (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)})
128, 10, 11syl2an 598 . . . . . . . . 9 (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥𝑊) → {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
135, 12eqtrd 2833 . . . . . . . 8 (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥𝑊) → {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
1413eqeq2d 2809 . . . . . . 7 (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑈 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
156adantl 485 . . . . . . . . . . . 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 11902 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0𝑁 ≠ 0))
18 eqeq1 2802 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 = (♯‘𝑥) → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
1918eqcoms 2806 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑥) = 𝑁 → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
20 hasheq0 13723 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ Word (Vtx‘𝐺) → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
2119, 20sylan9bbr 514 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 = 0 ↔ 𝑥 = ∅))
2221necon3bid 3031 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ≠ 0 ↔ 𝑥 ≠ ∅))
2322biimpcd 252 . . . . . . . . . . . . . . . . . 18 (𝑁 ≠ 0 → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
2417, 23simplbiim 508 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
2524impcom 411 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ≠ ∅)
26 simplr 768 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (♯‘𝑥) = 𝑁)
2726eqcomd 2804 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑁 = (♯‘𝑥))
2816, 25, 273jca 1125 . . . . . . . . . . . . . . 15 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)))
2928ex 416 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ∈ ℕ → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
30 eqid 2798 . . . . . . . . . . . . . . 15 (Vtx‘𝐺) = (Vtx‘𝐺)
3130clwwlknbp 27830 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
3229, 31syl11 33 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
339, 32syl5bi 245 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → (𝑥𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
3415, 33syl 17 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑥𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
3534imp 410 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)))
36 scshwfzeqfzo 14182 . . . . . . . . . 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)})
3735, 36syl 17 . . . . . . . . 9 (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥𝑊) → {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
3837eqeq2d 2809 . . . . . . . 8 (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (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 6646 . . . . . . . . . . . . . . 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 16033 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑥) ∈ ℙ → (♯‘𝑥) ∈ (ℤ‘2))
4241adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ) → (♯‘𝑥) ∈ (ℤ‘2))
4342adantl 485 . . . . . . . . . . . . . . . . 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 27860 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ (ℤ‘2) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) → ∃𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) ≠ (𝑥‘0))
4640, 43, 44, 45syl3anc 1368 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)) → ∃𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) ≠ (𝑥‘0))
47 oveq2 7144 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
4847eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
4948cbvrexvw 3397 . . . . . . . . . . . . . . . . . . . 20 (∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
50 eqeq1 2802 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ 𝑢 = (𝑥 cyclShift 𝑚)))
51 eqcom 2805 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢)
5250, 51syl6bb 290 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢))
5352rexbidv 3256 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑢 → (∃𝑚 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
5449, 53syl5bb 286 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑢 → (∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
5554cbvrabv 3439 . . . . . . . . . . . . . . . . . 18 {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} = {𝑢 ∈ Word (Vtx‘𝐺) ∣ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢}
5655cshwshashnsame 16432 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) → (∃𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) ≠ (𝑥‘0) → (♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥)))
5756ad2ant2rl 748 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)) ∧ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)) → (∃𝑖 ∈ (0..^(♯‘𝑥))(𝑥𝑖) ≠ (𝑥‘0) → (♯‘{𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥)))
5846, 57mpd 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))
5939, 58sylan9eqr 2855 . . . . . . . . . . . . . 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))
6059exp41 438 . . . . . . . . . . . . 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)))))
6160adantr 484 . . . . . . . . . . . 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 7143 . . . . . . . . . . . . . . . 16 (𝑁 = (♯‘𝑥) → (𝑁 ClWWalksN 𝐺) = ((♯‘𝑥) ClWWalksN 𝐺))
6362eleq2d 2875 . . . . . . . . . . . . . . 15 (𝑁 = (♯‘𝑥) → (𝑥 ∈ (𝑁 ClWWalksN 𝐺) ↔ 𝑥 ∈ ((♯‘𝑥) ClWWalksN 𝐺)))
64 eleq1 2877 . . . . . . . . . . . . . . . . 17 (𝑁 = (♯‘𝑥) → (𝑁 ∈ ℙ ↔ (♯‘𝑥) ∈ ℙ))
6564anbi2d 631 . . . . . . . . . . . . . . . 16 (𝑁 = (♯‘𝑥) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ↔ (𝐺 ∈ UMGraph ∧ (♯‘𝑥) ∈ ℙ)))
66 oveq2 7144 . . . . . . . . . . . . . . . . . . . 20 (𝑁 = (♯‘𝑥) → (0..^𝑁) = (0..^(♯‘𝑥)))
6766rexeqdv 3365 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (♯‘𝑥) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
6867rabbidv 3427 . . . . . . . . . . . . . . . . . 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)})
6968eqeq2d 2809 . . . . . . . . . . . . . . . . 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 2810 . . . . . . . . . . . . . . . . 17 (𝑁 = (♯‘𝑥) → ((♯‘𝑈) = 𝑁 ↔ (♯‘𝑈) = (♯‘𝑥)))
7169, 70imbi12d 348 . . . . . . . . . . . . . . . 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))))
7265, 71imbi12d 348 . . . . . . . . . . . . . . 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)))))
7363, 72imbi12d 348 . . . . . . . . . . . . . 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))))))
7473eqcoms 2806 . . . . . . . . . . . . 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))))))
7574adantl 485 . . . . . . . . . . . 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))))))
7661, 75mpbird 260 . . . . . . . . . . 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))))
7731, 76mpcom 38 . . . . . . . . . 10 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁)))
7877, 1eleq2s 2908 . . . . . . . . 9 (𝑥𝑊 → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁)))
7978impcom 411 . . . . . . . 8 (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁))
8038, 79sylbid 243 . . . . . . 7 (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑈 = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁))
8114, 80sylbid 243 . . . . . 6 (((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑈 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁))
8281rexlimdva 3243 . . . . 5 ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (∃𝑥𝑊 𝑈 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (♯‘𝑈) = 𝑁))
8382com12 32 . . . 4 (∃𝑥𝑊 𝑈 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (♯‘𝑈) = 𝑁))
843, 83syl6bi 256 . . 3 (𝑈 ∈ (𝑊 / ) → (𝑈 ∈ (𝑊 / ) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (♯‘𝑈) = 𝑁)))
8584pm2.43i 52 . 2 (𝑈 ∈ (𝑊 / ) → ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (♯‘𝑈) = 𝑁))
8685com12 32 1 ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ) → (♯‘𝑈) = 𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  {crab 3110  ∅c0 4243  {copab 5093  ‘cfv 6325  (class class class)co 7136   / cqs 8274  0cc0 10529  ℕcn 11628  2c2 11683  ℕ0cn0 11888  ℤ≥cuz 12234  ...cfz 12888  ..^cfzo 13031  ♯chash 13689  Word cword 13860   cyclShift ccsh 14144  ℙcprime 16008  Vtxcvtx 26799  UMGraphcumgr 26884   ClWWalksN cclwwlkn 27819 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-inf2 9091  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-disj 4997  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-2o 8089  df-oadd 8092  df-er 8275  df-ec 8277  df-qs 8281  df-map 8394  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-sup 8893  df-inf 8894  df-oi 8961  df-dju 9317  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11629  df-2 11691  df-3 11692  df-n0 11889  df-xnn0 11959  df-z 11973  df-uz 12235  df-rp 12381  df-ico 12735  df-fz 12889  df-fzo 13032  df-fl 13160  df-mod 13236  df-seq 13368  df-exp 13429  df-hash 13690  df-word 13861  df-lsw 13909  df-concat 13917  df-substr 13997  df-pfx 14027  df-reps 14125  df-csh 14145  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-sum 15038  df-dvds 15603  df-gcd 15837  df-prm 16009  df-phi 16096  df-edg 26851  df-umgr 26886  df-clwwlk 27777  df-clwwlkn 27820 This theorem is referenced by:  fusgrhashclwwlkn  27874
 Copyright terms: Public domain W3C validator