Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlk.r |
⊢ ∼ = { 〈 𝑢 , 𝑤 〉 ∣ ( 𝑢 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 𝑢 = ( 𝑤 cyclShift 𝑛 ) ) } |
2 |
|
anidm |
⊢ ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ) ↔ 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ) |
3 |
2
|
anbi1i |
⊢ ( ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ↔ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
4 |
|
df-3an |
⊢ ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ↔ ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
5 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
6 |
5
|
clwwlkbp |
⊢ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑥 ≠ ∅ ) ) |
7 |
|
cshw0 |
⊢ ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) → ( 𝑥 cyclShift 0 ) = 𝑥 ) |
8 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
9 |
8
|
a1i |
⊢ ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) → 0 ∈ ℕ0 ) |
10 |
|
lencl |
⊢ ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) → ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) |
11 |
|
hashge0 |
⊢ ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) → 0 ≤ ( ♯ ‘ 𝑥 ) ) |
12 |
|
elfz2nn0 |
⊢ ( 0 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ↔ ( 0 ∈ ℕ0 ∧ ( ♯ ‘ 𝑥 ) ∈ ℕ0 ∧ 0 ≤ ( ♯ ‘ 𝑥 ) ) ) |
13 |
9 10 11 12
|
syl3anbrc |
⊢ ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) → 0 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) |
14 |
|
eqcom |
⊢ ( ( 𝑥 cyclShift 0 ) = 𝑥 ↔ 𝑥 = ( 𝑥 cyclShift 0 ) ) |
15 |
14
|
biimpi |
⊢ ( ( 𝑥 cyclShift 0 ) = 𝑥 → 𝑥 = ( 𝑥 cyclShift 0 ) ) |
16 |
|
oveq2 |
⊢ ( 𝑛 = 0 → ( 𝑥 cyclShift 𝑛 ) = ( 𝑥 cyclShift 0 ) ) |
17 |
16
|
rspceeqv |
⊢ ( ( 0 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ∧ 𝑥 = ( 𝑥 cyclShift 0 ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) |
18 |
13 15 17
|
syl2an |
⊢ ( ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( 𝑥 cyclShift 0 ) = 𝑥 ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) |
19 |
7 18
|
mpdan |
⊢ ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) |
20 |
19
|
3ad2ant2 |
⊢ ( ( 𝐺 ∈ V ∧ 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑥 ≠ ∅ ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) |
21 |
6 20
|
syl |
⊢ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) |
22 |
21
|
pm4.71i |
⊢ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ↔ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
23 |
3 4 22
|
3bitr4ri |
⊢ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ↔ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
24 |
1
|
erclwwlkeq |
⊢ ( ( 𝑥 ∈ V ∧ 𝑥 ∈ V ) → ( 𝑥 ∼ 𝑥 ↔ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) ) |
25 |
24
|
el2v |
⊢ ( 𝑥 ∼ 𝑥 ↔ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) 𝑥 = ( 𝑥 cyclShift 𝑛 ) ) ) |
26 |
23 25
|
bitr4i |
⊢ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ↔ 𝑥 ∼ 𝑥 ) |