Step | Hyp | Ref
| Expression |
1 | | erclwwlkn.w |
. . . . 5
⊢ 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁) |
2 | | erclwwlkn.r |
. . . . 5
⊢ ∼ =
{〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} |
3 | 1, 2 | eclclwwlkn1 26359 |
. . . 4
⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) ↔
∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})) |
4 | | rabeq 3166 |
. . . . . . . . . 10
⊢ (𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁) → {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) |
5 | 1, 4 | mp1i 13 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) |
6 | | prmnn 15226 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℙ → 𝑁 ∈
ℕ) |
7 | 6 | nnnn0d 11228 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℙ → 𝑁 ∈
ℕ0) |
8 | 1 | eleq2i 2680 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) |
9 | 8 | biimpi 205 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑊 → 𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) |
10 | | clwwlknscsh 26347 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ ((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)}) |
11 | 7, 9, 10 | syl2an 493 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) |
12 | 5, 11 | eqtrd 2644 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) |
13 | 12 | eqeq2d 2620 |
. . . . . . 7
⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})) |
14 | | clwwlknprop 26300 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁))) |
15 | | simpll 786 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) ∧ 𝑁 ∈ ℕ) → 𝑥 ∈ Word 𝑉) |
16 | | elnnne0 11183 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0
∧ 𝑁 ≠
0)) |
17 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑁 = (#‘𝑥) → (𝑁 = 0 ↔ (#‘𝑥) = 0)) |
18 | 17 | eqcoms 2618 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((#‘𝑥) = 𝑁 → (𝑁 = 0 ↔ (#‘𝑥) = 0)) |
19 | 18 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 ∈ ℕ0
∧ (#‘𝑥) = 𝑁) → (𝑁 = 0 ↔ (#‘𝑥) = 0)) |
20 | | hasheq0 13015 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ∈ Word 𝑉 → ((#‘𝑥) = 0 ↔ 𝑥 = ∅)) |
21 | 19, 20 | sylan9bbr 733 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) → (𝑁 = 0 ↔ 𝑥 = ∅)) |
22 | 21 | necon3bid 2826 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) → (𝑁 ≠ 0 ↔ 𝑥 ≠ ∅)) |
23 | 22 | biimpcd 238 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ≠ 0 → ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) → 𝑥 ≠ ∅)) |
24 | 23 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈ ℕ0
∧ 𝑁 ≠ 0) →
((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) → 𝑥 ≠ ∅)) |
25 | 16, 24 | sylbi 206 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ → ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) → 𝑥 ≠ ∅)) |
26 | 25 | impcom 445 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) ∧ 𝑁 ∈ ℕ) → 𝑥 ≠ ∅) |
27 | | eqcom 2617 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((#‘𝑥) = 𝑁 ↔ 𝑁 = (#‘𝑥)) |
28 | 27 | biimpi 205 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((#‘𝑥) = 𝑁 → 𝑁 = (#‘𝑥)) |
29 | 28 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℕ0
∧ (#‘𝑥) = 𝑁) → 𝑁 = (#‘𝑥)) |
30 | 29 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) ∧ 𝑁 ∈ ℕ) → 𝑁 = (#‘𝑥)) |
31 | 15, 26, 30 | 3jca 1235 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ Word 𝑉 ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))) |
32 | 31 | ex 449 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) → (𝑁 ∈ ℕ → (𝑥 ∈ Word 𝑉 ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)))) |
33 | 32 | 3adant1 1072 |
. . . . . . . . . . . . . . 15
⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) → (𝑁 ∈ ℕ → (𝑥 ∈ Word 𝑉 ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)))) |
34 | 14, 33 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ → (𝑥 ∈ Word 𝑉 ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)))) |
35 | 34 | com12 32 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ → (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑥 ∈ Word 𝑉 ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)))) |
36 | 8, 35 | syl5bi 231 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word 𝑉 ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)))) |
37 | 6, 36 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℙ → (𝑥 ∈ 𝑊 → (𝑥 ∈ Word 𝑉 ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)))) |
38 | 37 | imp 444 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word 𝑉 ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥))) |
39 | | scshwfzeqfzo 13423 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ Word 𝑉 ∧ 𝑥 ≠ ∅ ∧ 𝑁 = (#‘𝑥)) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) |
40 | 38, 39 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) |
41 | 40 | eqeq2d 2620 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)})) |
42 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚)) |
43 | 42 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚))) |
44 | 43 | cbvrexv 3148 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∃𝑛 ∈
(0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)) |
45 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ 𝑢 = (𝑥 cyclShift 𝑚))) |
46 | | eqcom 2617 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑢 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢) |
47 | 45, 46 | syl6bb 275 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑢 → (𝑦 = (𝑥 cyclShift 𝑚) ↔ (𝑥 cyclShift 𝑚) = 𝑢)) |
48 | 47 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑢 → (∃𝑚 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0..^(#‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢)) |
49 | 44, 48 | syl5bb 271 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑢 → (∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0..^(#‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢)) |
50 | 49 | cbvrabv 3172 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} = {𝑢 ∈ Word 𝑉 ∣ ∃𝑚 ∈ (0..^(#‘𝑥))(𝑥 cyclShift 𝑚) = 𝑢} |
51 | 50 | cshwshash 15649 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) ∈ ℙ) → ((#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥) ∨ (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1)) |
52 | 51 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) → ((#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥) ∨ (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1)) |
53 | 52 | orcomd 402 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) → ((#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1 ∨ (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥))) |
54 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (#‘𝑈) = (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)})) |
55 | 54 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ↔ (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1)) |
56 | 54 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = (#‘𝑥) ↔ (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥))) |
57 | 55, 56 | orbi12d 742 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → (((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥)) ↔ ((#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1 ∨ (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥)))) |
58 | 57 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) → (((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥)) ↔ ((#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = 1 ∨ (#‘{𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) = (#‘𝑥)))) |
59 | 53, 58 | mpbird 246 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) ∈ ℙ) ∧ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) → ((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥))) |
60 | 59 | ex 449 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) ∈ ℙ) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥)))) |
61 | 60 | ex 449 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ Word 𝑉 → ((#‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥))))) |
62 | 61 | 3ad2ant2 1076 |
. . . . . . . . . . . 12
⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) → ((#‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥))))) |
63 | | eleq1 2676 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 = (#‘𝑥) → (𝑁 ∈ ℙ ↔ (#‘𝑥) ∈
ℙ)) |
64 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 = (#‘𝑥) → (0..^𝑁) = (0..^(#‘𝑥))) |
65 | 64 | rexeqdv 3122 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 = (#‘𝑥) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))) |
66 | 65 | rabbidv 3164 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 = (#‘𝑥) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)}) |
67 | 66 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 = (#‘𝑥) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} ↔ 𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)})) |
68 | | eqeq2 2621 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 = (#‘𝑥) → ((#‘𝑈) = 𝑁 ↔ (#‘𝑈) = (#‘𝑥))) |
69 | 68 | orbi2d 734 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 = (#‘𝑥) → (((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁) ↔ ((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥)))) |
70 | 67, 69 | imbi12d 333 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 = (#‘𝑥) → ((𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁)) ↔ (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥))))) |
71 | 63, 70 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 = (#‘𝑥) → ((𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))) ↔ ((#‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥)))))) |
72 | 71 | eqcoms 2618 |
. . . . . . . . . . . . . 14
⊢
((#‘𝑥) = 𝑁 → ((𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))) ↔ ((#‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥)))))) |
73 | 72 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ0
∧ (#‘𝑥) = 𝑁) → ((𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))) ↔ ((#‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥)))))) |
74 | 73 | 3ad2ant3 1077 |
. . . . . . . . . . . 12
⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) → ((𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))) ↔ ((#‘𝑥) ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = (#‘𝑥)))))) |
75 | 62, 74 | mpbird 246 |
. . . . . . . . . . 11
⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧
(#‘𝑥) = 𝑁)) → (𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁)))) |
76 | 14, 75 | syl 17 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁)))) |
77 | 8, 76 | sylbi 206 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝑊 → (𝑁 ∈ ℙ → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁)))) |
78 | 77 | impcom 445 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))) |
79 | 41, 78 | sylbid 229 |
. . . . . . 7
⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))) |
80 | 13, 79 | sylbid 229 |
. . . . . 6
⊢ ((𝑁 ∈ ℙ ∧ 𝑥 ∈ 𝑊) → (𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))) |
81 | 80 | rexlimdva 3013 |
. . . . 5
⊢ (𝑁 ∈ ℙ →
(∃𝑥 ∈ 𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))) |
82 | 81 | com12 32 |
. . . 4
⊢
(∃𝑥 ∈
𝑊 𝑈 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} → (𝑁 ∈ ℙ → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))) |
83 | 3, 82 | syl6bi 242 |
. . 3
⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑈 ∈ (𝑊 / ∼ ) → (𝑁 ∈ ℙ →
((#‘𝑈) = 1 ∨
(#‘𝑈) = 𝑁)))) |
84 | 83 | pm2.43i 50 |
. 2
⊢ (𝑈 ∈ (𝑊 / ∼ ) → (𝑁 ∈ ℙ →
((#‘𝑈) = 1 ∨
(#‘𝑈) = 𝑁))) |
85 | 84 | impcom 445 |
1
⊢ ((𝑁 ∈ ℙ ∧ 𝑈 ∈ (𝑊 / ∼ )) →
((#‘𝑈) = 1 ∨
(#‘𝑈) = 𝑁)) |