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Theorem hashecclwwlkn1 29874
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 𝑊 𝑚 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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
31, 2eclclwwlkn1 29872 . . . 4 (𝑈 ∈ (𝑊 / ∌ ) → (𝑈 ∈ (𝑊 / ∌ ) ↔ ∃𝑥 ∈ 𝑊 𝑈 = {𝑊 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)}))
4 rabeq 3441 . . . . . . . . . 10 (𝑊 = (𝑁 ClWWalksN 𝐺) → {𝑊 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} = {𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)})
51, 4mp1i 13 . . . . . . . . 9 ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑊 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} = {𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)})
6 prmnn 16636 . . . . . . . . . . 11 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
76nnnn0d 12554 . . . . . . . . . 10 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ0)
81eleq2i 2820 . . . . . . . . . . 11 (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ (𝑁 ClWWalksN 𝐺))
98biimpi 215 . . . . . . . . . 10 (𝑥 ∈ 𝑊 → 𝑥 ∈ (𝑁 ClWWalksN 𝐺))
10 clwwlknscsh 29859 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)})
117, 9, 10syl2an 595 . . . . . . . . 9 ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)})
125, 11eqtrd 2767 . . . . . . . 8 ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑊 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)})
1312eqeq2d 2738 . . . . . . 7 ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑊 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)}))
14 simpll 766 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ∈ Word (Vtx‘𝐺))
15 elnnne0 12508 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
16 eqeq1 2731 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 = (♯‘𝑥) → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
1716eqcoms 2735 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑥) = 𝑁 → (𝑁 = 0 ↔ (♯‘𝑥) = 0))
18 hasheq0 14346 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ Word (Vtx‘𝐺) → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
1917, 18sylan9bbr 510 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 = 0 ↔ 𝑥 = ∅))
2019necon3bid 2980 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ≠ 0 ↔ 𝑥 ≠ ∅))
2120biimpcd 248 . . . . . . . . . . . . . . . . . 18 (𝑁 ≠ 0 → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
2215, 21simplbiim 504 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → 𝑥 ≠ ∅))
2322impcom 407 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑥 ≠ ∅)
24 simplr 768 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (♯‘𝑥) = 𝑁)
2524eqcomd 2733 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑁 = (♯‘𝑥))
2614, 23, 253jca 1126 . . . . . . . . . . . . . . 15 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)))
2726ex 412 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ∈ ℕ → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
28 eqid 2727 . . . . . . . . . . . . . . 15 (Vtx‘𝐺) = (Vtx‘𝐺)
2928clwwlknbp 29832 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
3027, 29syl11 33 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
318, 30biimtrid 241 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
326, 31syl 17 . . . . . . . . . . 11 (𝑁 ∈ ℙ → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥))))
3332imp 406 . . . . . . . . . 10 ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)))
34 scshwfzeqfzo 14801 . . . . . . . . . 10 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (♯‘𝑥)) → {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)})
3533, 34syl 17 . . . . . . . . 9 ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)})
3635eqeq2d 2738 . . . . . . . 8 ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)}))
37 oveq2 7422 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
3837eqeq2d 2738 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (𝑊 = (𝑥 cyclShift 𝑛) ↔ 𝑊 = (𝑥 cyclShift 𝑚)))
3938cbvrexvw 3230 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑚))
40 eqeq1 2731 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑊 = 𝑢 → (𝑊 = (𝑥 cyclShift 𝑚) ↔ 𝑢 = (𝑥 cyclShift 𝑚)))
41 eqcom 2734 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢)
4240, 41bitrdi 287 . . . . . . . . . . . . . . . . . . . . . 22 (𝑊 = 𝑢 → (𝑊 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢))
4342rexbidv 3173 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 = 𝑢 → (∃𝑚 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
4439, 43bitrid 283 . . . . . . . . . . . . . . . . . . . 20 (𝑊 = 𝑢 → (∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
4544cbvrabv 3437 . . . . . . . . . . . . . . . . . . 19 {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} = {𝑢 ∈ Word (Vtx‘𝐺) ∣ ∃𝑚 ∈ (0..^(♯‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢}
4645cshwshash 17065 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) → ((♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥) √ (♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = 1))
4746adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) → ((♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥) √ (♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = 1))
4847orcomd 870 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) → ((♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = 1 √ (♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥)))
49 fveqeq2 6900 . . . . . . . . . . . . . . . . . 18 (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 ↔ (♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = 1))
50 fveqeq2 6900 . . . . . . . . . . . . . . . . . 18 (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = (♯‘𝑥) ↔ (♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥)))
5149, 50orbi12d 917 . . . . . . . . . . . . . . . . 17 (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} → (((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥)) ↔ ((♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = 1 √ (♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥))))
5251adantl 481 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) → (((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥)) ↔ ((♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = 1 √ (♯‘{𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) = (♯‘𝑥))))
5348, 52mpbird 257 . . . . . . . . . . . . . . 15 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}) → ((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥)))
5453ex 412 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) ∈ ℙ) → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥))))
5554ex 412 . . . . . . . . . . . . 13 (𝑥 ∈ Word (Vtx‘𝐺) → ((♯‘𝑥) ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥)))))
5655adantr 480 . . . . . . . . . . . 12 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → ((♯‘𝑥) ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥)))))
57 eleq1 2816 . . . . . . . . . . . . . . 15 (𝑁 = (♯‘𝑥) → (𝑁 ∈ ℙ ↔ (♯‘𝑥) ∈ ℙ))
58 oveq2 7422 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (♯‘𝑥) → (0..^𝑁) = (0..^(♯‘𝑥)))
5958rexeqdv 3321 . . . . . . . . . . . . . . . . . 18 (𝑁 = (♯‘𝑥) → (∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)))
6059rabbidv 3435 . . . . . . . . . . . . . . . . 17 (𝑁 = (♯‘𝑥) → {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)} = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)})
6160eqeq2d 2738 . . . . . . . . . . . . . . . 16 (𝑁 = (♯‘𝑥) → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)}))
62 eqeq2 2739 . . . . . . . . . . . . . . . . 17 (𝑁 = (♯‘𝑥) → ((♯‘𝑈) = 𝑁 ↔ (♯‘𝑈) = (♯‘𝑥)))
6362orbi2d 914 . . . . . . . . . . . . . . . 16 (𝑁 = (♯‘𝑥) → (((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁) ↔ ((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥))))
6461, 63imbi12d 344 . . . . . . . . . . . . . . 15 (𝑁 = (♯‘𝑥) → ((𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁)) ↔ (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥)))))
6557, 64imbi12d 344 . . . . . . . . . . . . . 14 (𝑁 = (♯‘𝑥) → ((𝑁 ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁))) ↔ ((♯‘𝑥) ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥))))))
6665eqcoms 2735 . . . . . . . . . . . . 13 ((♯‘𝑥) = 𝑁 → ((𝑁 ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁))) ↔ ((♯‘𝑥) ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥))))))
6766adantl 481 . . . . . . . . . . . 12 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → ((𝑁 ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁))) ↔ ((♯‘𝑥) ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = (♯‘𝑥))))))
6856, 67mpbird 257 . . . . . . . . . . 11 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑁 ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁))))
6929, 68syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑁 ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁))))
7069, 1eleq2s 2846 . . . . . . . . 9 (𝑥 ∈ 𝑊 → (𝑁 ∈ ℙ → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁))))
7170impcom 407 . . . . . . . 8 ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0..^𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁)))
7236, 71sylbid 239 . . . . . . 7 ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁)))
7313, 72sylbid 239 . . . . . 6 ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑊 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁)))
7473rexlimdva 3150 . . . . 5 (𝑁 ∈ ℙ → (∃𝑥 ∈ 𝑊 𝑈 = {𝑊 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁)))
7574com12 32 . . . 4 (∃𝑥 ∈ 𝑊 𝑈 = {𝑊 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)} → (𝑁 ∈ ℙ → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁)))
763, 75biimtrdi 252 . . 3 (𝑈 ∈ (𝑊 / ∌ ) → (𝑈 ∈ (𝑊 / ∌ ) → (𝑁 ∈ ℙ → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁))))
7776pm2.43i 52 . 2 (𝑈 ∈ (𝑊 / ∌ ) → (𝑁 ∈ ℙ → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁)))
7877impcom 407 1 ((𝑁 ∈ ℙ ∧ 𝑈 ∈ (𝑊 / ∌ )) → ((♯‘𝑈) = 1 √ (♯‘𝑈) = 𝑁))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   √ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  âˆƒwrex 3065  {crab 3427  âˆ…c0 4318  {copab 5204  â€˜cfv 6542  (class class class)co 7414   / cqs 8717  0cc0 11130  1c1 11131  â„•cn 12234  â„•0cn0 12494  ...cfz 13508  ..^cfzo 13651  â™¯chash 14313  Word cword 14488   cyclShift ccsh 14762  â„™cprime 16633  Vtxcvtx 28796   ClWWalksN cclwwlkn 29821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-ec 8720  df-qs 8724  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-rp 12999  df-ico 13354  df-fz 13509  df-fzo 13652  df-fl 13781  df-mod 13859  df-seq 13991  df-exp 14051  df-hash 14314  df-word 14489  df-lsw 14537  df-concat 14545  df-substr 14615  df-pfx 14645  df-reps 14743  df-csh 14763  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657  df-dvds 16223  df-gcd 16461  df-prm 16634  df-phi 16726  df-clwwlk 29779  df-clwwlkn 29822
This theorem is referenced by: (None)
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