Step | Hyp | Ref
| Expression |
1 | | vex 3203 |
. 2
⊢ 𝑥 ∈ V |
2 | | vex 3203 |
. 2
⊢ 𝑦 ∈ V |
3 | | erclwwlks.r |
. . . 4
⊢ ∼ =
{〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} |
4 | 3 | erclwwlkseqlen 26933 |
. . 3
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (#‘𝑥) = (#‘𝑦))) |
5 | 3 | erclwwlkseq 26932 |
. . . 4
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)))) |
6 | | simpl2 1065 |
. . . . . . 7
⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 ∈ (ClWWalks‘𝐺)) |
7 | | simpl1 1064 |
. . . . . . 7
⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑥 ∈ (ClWWalks‘𝐺)) |
8 | | eqid 2622 |
. . . . . . . . . . . . . . . . . 18
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
9 | 8 | clwwlkbp 26883 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ≠ ∅)) |
10 | 9 | simp2d 1074 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ (ClWWalks‘𝐺) → 𝑦 ∈ Word (Vtx‘𝐺)) |
11 | 10 | ad2antlr 763 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺)) |
12 | | simpr 477 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (#‘𝑥) = (#‘𝑦)) → (#‘𝑥) = (#‘𝑦)) |
13 | 11, 12 | cshwcshid 13573 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑛 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))) |
14 | 13 | expd 452 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (#‘𝑥) = (#‘𝑦)) → (𝑛 ∈ (0...(#‘𝑦)) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))) |
15 | 14 | rexlimdv 3030 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))) |
16 | 15 | ex 450 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ((#‘𝑥) = (#‘𝑦) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))) |
17 | 16 | com23 86 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((#‘𝑥) = (#‘𝑦) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))) |
18 | 17 | 3impia 1261 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑥) = (#‘𝑦) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))) |
19 | 18 | imp 445 |
. . . . . . . 8
⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)) |
20 | | oveq2 6658 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚)) |
21 | 20 | eqeq2d 2632 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚))) |
22 | 21 | cbvrexv 3172 |
. . . . . . . 8
⊢
(∃𝑛 ∈
(0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)) |
23 | 19, 22 | sylibr 224 |
. . . . . . 7
⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)) |
24 | 6, 7, 23 | 3jca 1242 |
. . . . . 6
⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))) |
25 | 3 | erclwwlkseq 26932 |
. . . . . . 7
⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))) |
26 | 25 | ancoms 469 |
. . . . . 6
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))) |
27 | 24, 26 | syl5ibr 236 |
. . . . 5
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 ∼ 𝑥)) |
28 | 27 | expd 452 |
. . . 4
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑥) = (#‘𝑦) → 𝑦 ∼ 𝑥))) |
29 | 5, 28 | sylbid 230 |
. . 3
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → ((#‘𝑥) = (#‘𝑦) → 𝑦 ∼ 𝑥))) |
30 | 4, 29 | mpdd 43 |
. 2
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)) |
31 | 1, 2, 30 | mp2an 708 |
1
⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥) |