Step | Hyp | Ref
| Expression |
1 | | vex 3176 |
. 2
⊢ 𝑥 ∈ V |
2 | | vex 3176 |
. 2
⊢ 𝑦 ∈ V |
3 | | erclwwlksn.w |
. . . 4
⊢ 𝑊 = (𝑁 ClWWalkSN 𝐺) |
4 | | erclwwlksn.r |
. . . 4
⊢ ∼ =
{〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} |
5 | 3, 4 | erclwwlksneqlen 41252 |
. . 3
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (#‘𝑥) = (#‘𝑦))) |
6 | 3, 4 | erclwwlksneq 41251 |
. . . 4
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)))) |
7 | | simpl2 1058 |
. . . . . . 7
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 ∈ 𝑊) |
8 | | simpl1 1057 |
. . . . . . 7
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑥 ∈ 𝑊) |
9 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
10 | 9 | clwwlknbp 41193 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁)) |
11 | | eqcom 2617 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((#‘𝑥) = 𝑁 ↔ 𝑁 = (#‘𝑥)) |
12 | 11 | biimpi 205 |
. . . . . . . . . . . . . . . . . . 19
⊢
((#‘𝑥) = 𝑁 → 𝑁 = (#‘𝑥)) |
13 | 10, 12 | simpl2im 656 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → 𝑁 = (#‘𝑥)) |
14 | 13, 3 | eleq2s 2706 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ 𝑊 → 𝑁 = (#‘𝑥)) |
15 | 14 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → 𝑁 = (#‘𝑥)) |
16 | 15 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑁 = (#‘𝑥)) |
17 | 9 | clwwlksnwrd 41194 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (𝑁 ClWWalkSN 𝐺) → 𝑦 ∈ Word (Vtx‘𝐺)) |
18 | 17, 3 | eleq2s 2706 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ 𝑊 → 𝑦 ∈ Word (Vtx‘𝐺)) |
19 | 18 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → 𝑦 ∈ Word (Vtx‘𝐺)) |
20 | 19 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺)) |
21 | 20 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 = (#‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → 𝑦 ∈ Word (Vtx‘𝐺)) |
22 | | simprr 792 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 = (#‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (#‘𝑥) = (#‘𝑦)) |
23 | 21, 22 | cshwcshid 13424 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = (#‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → ((𝑛 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))) |
24 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 = (#‘𝑥) → (0...𝑁) = (0...(#‘𝑥))) |
25 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((#‘𝑥) =
(#‘𝑦) →
(0...(#‘𝑥)) =
(0...(#‘𝑦))) |
26 | 25 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → (0...(#‘𝑥)) = (0...(#‘𝑦))) |
27 | 24, 26 | sylan9eq 2664 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 = (#‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (0...𝑁) = (0...(#‘𝑦))) |
28 | 27 | eleq2d 2673 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 = (#‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑦)))) |
29 | 28 | anbi1d 737 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = (#‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) ↔ (𝑛 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)))) |
30 | 24 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 = (#‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (0...𝑁) = (0...(#‘𝑥))) |
31 | 30 | rexeqdv 3122 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = (#‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))) |
32 | 23, 29, 31 | 3imtr4d 282 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = (#‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))) |
33 | 16, 32 | mpancom 700 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))) |
34 | 33 | expd 451 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → (𝑛 ∈ (0...𝑁) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))) |
35 | 34 | rexlimdv 3012 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))) |
36 | 35 | ex 449 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → ((#‘𝑥) = (#‘𝑦) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))) |
37 | 36 | com23 84 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((#‘𝑥) = (#‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))) |
38 | 37 | 3impia 1253 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑥) = (#‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))) |
39 | 38 | imp 444 |
. . . . . . . 8
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)) |
40 | | oveq2 6557 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚)) |
41 | 40 | eqeq2d 2620 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚))) |
42 | 41 | cbvrexv 3148 |
. . . . . . . 8
⊢
(∃𝑛 ∈
(0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)) |
43 | 39, 42 | sylibr 223 |
. . . . . . 7
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)) |
44 | 7, 8, 43 | 3jca 1235 |
. . . . . 6
⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))) |
45 | 3, 4 | erclwwlksneq 41251 |
. . . . . . 7
⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))) |
46 | 45 | ancoms 468 |
. . . . . 6
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))) |
47 | 44, 46 | syl5ibr 235 |
. . . . 5
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 ∼ 𝑥)) |
48 | 47 | expd 451 |
. . . 4
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑥) = (#‘𝑦) → 𝑦 ∼ 𝑥))) |
49 | 6, 48 | sylbid 229 |
. . 3
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → ((#‘𝑥) = (#‘𝑦) → 𝑦 ∼ 𝑥))) |
50 | 5, 49 | mpdd 42 |
. 2
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)) |
51 | 1, 2, 50 | mp2an 704 |
1
⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥) |