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Theorem usghashecclwwlk 26362
 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.)
Hypotheses
Ref Expression
erclwwlkn.w 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
erclwwlkn.r = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
usghashecclwwlk ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ) → (#‘𝑈) = 𝑁))
Distinct variable groups:   𝑡,𝐸,𝑢   𝑡,𝑁,𝑢   𝑛,𝑉,𝑡,𝑢   𝑡,𝑊,𝑢   𝑛,𝑁   𝑛,𝑊   𝑛,𝐸   𝑈,𝑛,𝑢
Allowed substitution hints:   (𝑢,𝑡,𝑛)   𝑈(𝑡)

Proof of Theorem usghashecclwwlk
Dummy variables 𝑥 𝑦 𝑚 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 erclwwlkn.w . . . . 5 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
2 erclwwlkn.r . . . . 5 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
31, 2eclclwwlkn1 26359 . . . 4 (𝑈 ∈ (𝑊 / ) → (𝑈 ∈ (𝑊 / ) ↔ ∃𝑥𝑊 𝑈 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
4 rabeq 3166 . . . . . . . . . 10 (𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁) → {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
51, 4mp1i 13 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ∧ 𝑥𝑊) → {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
6 prmnn 15226 . . . . . . . . . . . 12 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
76nnnn0d 11228 . . . . . . . . . . 11 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ0)
87adantl 481 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → 𝑁 ∈ ℕ0)
91eleq2i 2680 . . . . . . . . . . 11 (𝑥𝑊𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
109biimpi 205 . . . . . . . . . 10 (𝑥𝑊𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
11 clwwlknscsh 26347 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
128, 10, 11syl2an 493 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ∧ 𝑥𝑊) → {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
135, 12eqtrd 2644 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ∧ 𝑥𝑊) → {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
1413eqeq2d 2620 . . . . . . 7 (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑈 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
156adantl 481 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → 𝑁 ∈ ℕ)
16 clwwlknprop 26300 . . . . . . . . . . . . . . 15 (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)))
17 simpll 786 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑁 ∈ ℕ) → 𝑥 ∈ Word 𝑉)
18 elnnne0 11183 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0𝑁 ≠ 0))
19 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (#‘𝑥) → (𝑁 = 0 ↔ (#‘𝑥) = 0))
2019eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝑥) = 𝑁 → (𝑁 = 0 ↔ (#‘𝑥) = 0))
2120adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁) → (𝑁 = 0 ↔ (#‘𝑥) = 0))
22 hasheq0 13015 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ Word 𝑉 → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
2321, 22sylan9bbr 733 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → (𝑁 = 0 ↔ 𝑥 = ∅))
2423necon3bid 2826 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → (𝑁 ≠ 0 ↔ 𝑥 ≠ ∅))
2524biimpcd 238 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ≠ 0 → ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → 𝑥 ≠ ∅))
2625adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝑁 ≠ 0) → ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → 𝑥 ≠ ∅))
2718, 26sylbi 206 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → 𝑥 ≠ ∅))
2827impcom 445 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑁 ∈ ℕ) → 𝑥 ≠ ∅)
29 eqcom 2617 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑥) = 𝑁𝑁 = (#‘𝑥))
3029biimpi 205 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑥) = 𝑁𝑁 = (#‘𝑥))
3130adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁) → 𝑁 = (#‘𝑥))
3231ad2antlr 759 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑁 ∈ ℕ) → 𝑁 = (#‘𝑥))
3317, 28, 323jca 1235 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ Word 𝑉𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)))
3433ex 449 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → (𝑁 ∈ ℕ → (𝑥 ∈ Word 𝑉𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
35343adant1 1072 . . . . . . . . . . . . . . 15 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → (𝑁 ∈ ℕ → (𝑥 ∈ Word 𝑉𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
3616, 35syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ → (𝑥 ∈ Word 𝑉𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
3736com12 32 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑥 ∈ Word 𝑉𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
389, 37syl5bi 231 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → (𝑥𝑊 → (𝑥 ∈ Word 𝑉𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
3915, 38syl 17 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑥𝑊 → (𝑥 ∈ Word 𝑉𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))))
4039imp 444 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑥 ∈ Word 𝑉𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)))
41 scshwfzeqfzo 13423 . . . . . . . . . 10 ((𝑥 ∈ Word 𝑉𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
4240, 41syl 17 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ∧ 𝑥𝑊) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
4342eqeq2d 2620 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
44 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}))
45 simprl 790 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ Word 𝑉𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥))) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ)) → 𝑉 USGrph 𝐸)
46 prmuz2 15246 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑥) ∈ ℙ → (#‘𝑥) ∈ (ℤ‘2))
4746adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ) → (#‘𝑥) ∈ (ℤ‘2))
4847adantl 481 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ Word 𝑉𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥))) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ)) → (#‘𝑥) ∈ (ℤ‘2))
49 simplr 788 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ Word 𝑉𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥))) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ)) → 𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥)))
50 usg2cwwkdifex 26349 . . . . . . . . . . . . . . . . 17 ((𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ (ℤ‘2) ∧ 𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥))) → ∃𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) ≠ (𝑥‘0))
5145, 48, 49, 50syl3anc 1318 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word 𝑉𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥))) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ)) → ∃𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) ≠ (𝑥‘0))
52 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
5352eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
5453cbvrexv 3148 . . . . . . . . . . . . . . . . . . . 20 (∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
55 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ 𝑢 = (𝑥 cyclShift 𝑚)))
56 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢)
5755, 56syl6bb 275 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢))
5857rexbidv 3034 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑢 → (∃𝑚 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0..^(#‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
5954, 58syl5bb 271 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑢 → (∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(#‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢))
6059cbvrabv 3172 . . . . . . . . . . . . . . . . . 18 {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} = {𝑢 ∈ Word 𝑉 ∣ ∃𝑚 ∈ (0..^(#‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢}
6160cshwshashnsame 15648 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) ∈ ℙ) → (∃𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) ≠ (𝑥‘0) → (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥)))
6261ad2ant2rl 781 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ Word 𝑉𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥))) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ)) → (∃𝑖 ∈ (0..^(#‘𝑥))(𝑥𝑖) ≠ (𝑥‘0) → (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥)))
6351, 62mpd 15 . . . . . . . . . . . . . . 15 (((𝑥 ∈ Word 𝑉𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥))) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ)) → (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥))
6444, 63sylan9eqr 2666 . . . . . . . . . . . . . 14 ((((𝑥 ∈ Word 𝑉𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥))) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ)) ∧ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) → (#‘𝑈) = (#‘𝑥))
6564exp41 636 . . . . . . . . . . . . 13 (𝑥 ∈ Word 𝑉 → (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥)) → ((𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥)))))
66653ad2ant2 1076 . . . . . . . . . . . 12 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥)) → ((𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥)))))
67 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑁 = (#‘𝑥) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥)))
6867eleq2d 2673 . . . . . . . . . . . . . . . 16 (𝑁 = (#‘𝑥) → (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ 𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥))))
69 eleq1 2676 . . . . . . . . . . . . . . . . . 18 (𝑁 = (#‘𝑥) → (𝑁 ∈ ℙ ↔ (#‘𝑥) ∈ ℙ))
7069anbi2d 736 . . . . . . . . . . . . . . . . 17 (𝑁 = (#‘𝑥) → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ↔ (𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ)))
71 oveq2 6557 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 = (#‘𝑥) → (0..^𝑁) = (0..^(#‘𝑥)))
7271rexeqdv 3122 . . . . . . . . . . . . . . . . . . . 20 (𝑁 = (#‘𝑥) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
7372rabbidv 3164 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (#‘𝑥) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)})
7473eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 (𝑁 = (#‘𝑥) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}))
75 eqeq2 2621 . . . . . . . . . . . . . . . . . 18 (𝑁 = (#‘𝑥) → ((#‘𝑈) = 𝑁 ↔ (#‘𝑈) = (#‘𝑥)))
7674, 75imbi12d 333 . . . . . . . . . . . . . . . . 17 (𝑁 = (#‘𝑥) → ((𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁) ↔ (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥))))
7770, 76imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑁 = (#‘𝑥) → (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁)) ↔ ((𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥)))))
7868, 77imbi12d 333 . . . . . . . . . . . . . . 15 (𝑁 = (#‘𝑥) → ((𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))) ↔ (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥)) → ((𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥))))))
7978eqcoms 2618 . . . . . . . . . . . . . 14 ((#‘𝑥) = 𝑁 → ((𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))) ↔ (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥)) → ((𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥))))))
8079adantl 481 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁) → ((𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))) ↔ (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥)) → ((𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥))))))
81803ad2ant3 1077 . . . . . . . . . . . 12 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → ((𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))) ↔ (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘(#‘𝑥)) → ((𝑉 USGrph 𝐸 ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘𝑥))))))
8266, 81mpbird 246 . . . . . . . . . . 11 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))))
8316, 82mpcom 37 . . . . . . . . . 10 (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁)))
849, 83sylbi 206 . . . . . . . . 9 (𝑥𝑊 → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁)))
8584impcom 445 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))
8643, 85sylbid 229 . . . . . . 7 (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))
8714, 86sylbid 229 . . . . . 6 (((𝑉 USGrph 𝐸𝑁 ∈ ℙ) ∧ 𝑥𝑊) → (𝑈 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))
8887rexlimdva 3013 . . . . 5 ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (∃𝑥𝑊 𝑈 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = 𝑁))
8988com12 32 . . . 4 (∃𝑥𝑊 𝑈 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (#‘𝑈) = 𝑁))
903, 89syl6bi 242 . . 3 (𝑈 ∈ (𝑊 / ) → (𝑈 ∈ (𝑊 / ) → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (#‘𝑈) = 𝑁)))
9190pm2.43i 50 . 2 (𝑈 ∈ (𝑊 / ) → ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (#‘𝑈) = 𝑁))
9291com12 32 1 ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ) → (#‘𝑈) = 𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173  ∅c0 3874   class class class wbr 4583  {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   USGrph cusg 25859   ClWWalksN cclwwlkn 26277 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-usgra 25862  df-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  hashclwwlkn  26363
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