Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlkn.w |
⊢ 𝑊 = ( 𝑁 ClWWalksN 𝐺 ) |
2 |
|
erclwwlkn.r |
⊢ ∼ = { 〈 𝑡 , 𝑢 〉 ∣ ( 𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) } |
3 |
|
df-3an |
⊢ ( ( 𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ↔ ( ( 𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
4 |
|
anidm |
⊢ ( ( 𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ↔ 𝑥 ∈ 𝑊 ) |
5 |
4
|
anbi1i |
⊢ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ↔ ( 𝑥 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
6 |
3 5
|
bitri |
⊢ ( ( 𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ↔ ( 𝑥 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
7 |
1 2
|
erclwwlkneq |
⊢ ( ( 𝑥 ∈ V ∧ 𝑥 ∈ V ) → ( 𝑥 ∼ 𝑥 ↔ ( 𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) ) |
8 |
7
|
el2v |
⊢ ( 𝑥 ∼ 𝑥 ↔ ( 𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
9 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
10 |
9
|
clwwlknwrd |
⊢ ( 𝑥 ∈ ( 𝑁 ClWWalksN 𝐺 ) → 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ) |
11 |
|
clwwlknnn |
⊢ ( 𝑥 ∈ ( 𝑁 ClWWalksN 𝐺 ) → 𝑁 ∈ ℕ ) |
12 |
|
cshw0 |
⊢ ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) → ( 𝑥 cyclShift 0 ) = 𝑥 ) |
13 |
|
nnnn0 |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 ) |
14 |
|
0elfz |
⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑁 ) ) |
15 |
13 14
|
syl |
⊢ ( 𝑁 ∈ ℕ → 0 ∈ ( 0 ... 𝑁 ) ) |
16 |
|
eqcom |
⊢ ( ( 𝑥 cyclShift 0 ) = 𝑥 ↔ 𝑥 = ( 𝑥 cyclShift 0 ) ) |
17 |
16
|
biimpi |
⊢ ( ( 𝑥 cyclShift 0 ) = 𝑥 → 𝑥 = ( 𝑥 cyclShift 0 ) ) |
18 |
|
oveq2 |
⊢ ( 𝑛 = 0 → ( 𝑥 cyclShift 𝑛 ) = ( 𝑥 cyclShift 0 ) ) |
19 |
18
|
rspceeqv |
⊢ ( ( 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑥 = ( 𝑥 cyclShift 0 ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) |
20 |
15 17 19
|
syl2anr |
⊢ ( ( ( 𝑥 cyclShift 0 ) = 𝑥 ∧ 𝑁 ∈ ℕ ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) |
21 |
20
|
ex |
⊢ ( ( 𝑥 cyclShift 0 ) = 𝑥 → ( 𝑁 ∈ ℕ → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
22 |
12 21
|
syl |
⊢ ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) → ( 𝑁 ∈ ℕ → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
23 |
10 11 22
|
sylc |
⊢ ( 𝑥 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) |
24 |
23 1
|
eleq2s |
⊢ ( 𝑥 ∈ 𝑊 → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) |
25 |
24
|
pm4.71i |
⊢ ( 𝑥 ∈ 𝑊 ↔ ( 𝑥 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
26 |
6 8 25
|
3bitr4ri |
⊢ ( 𝑥 ∈ 𝑊 ↔ 𝑥 ∼ 𝑥 ) |