Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlk.r |
⊢ ∼ = { 〈 𝑢 , 𝑤 〉 ∣ ( 𝑢 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 𝑢 = ( 𝑤 cyclShift 𝑛 ) ) } |
2 |
|
vex |
⊢ 𝑥 ∈ V |
3 |
|
vex |
⊢ 𝑦 ∈ V |
4 |
|
vex |
⊢ 𝑧 ∈ V |
5 |
1
|
erclwwlkeqlen |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) → ( 𝑥 ∼ 𝑦 → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) |
6 |
5
|
3adant3 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑦 → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) |
7 |
1
|
erclwwlkeqlen |
⊢ ( ( 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 → ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ) ) |
8 |
7
|
3adant1 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 → ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ) ) |
9 |
1
|
erclwwlkeq |
⊢ ( ( 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 ↔ ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
10 |
9
|
3adant1 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 ↔ ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
11 |
1
|
erclwwlkeq |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) → ( 𝑥 ∼ 𝑦 ↔ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) ) |
12 |
11
|
3adant3 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑦 ↔ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) ) |
13 |
|
simpr1 |
⊢ ( ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ∧ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) → 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ) |
14 |
|
simplr2 |
⊢ ( ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ∧ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) → 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ) |
15 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝑦 cyclShift 𝑛 ) = ( 𝑦 cyclShift 𝑚 ) ) |
16 |
15
|
eqeq2d |
⊢ ( 𝑛 = 𝑚 → ( 𝑥 = ( 𝑦 cyclShift 𝑛 ) ↔ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ) |
17 |
16
|
cbvrexvw |
⊢ ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ↔ ∃ 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) |
18 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝑧 cyclShift 𝑛 ) = ( 𝑧 cyclShift 𝑘 ) ) |
19 |
18
|
eqeq2d |
⊢ ( 𝑛 = 𝑘 → ( 𝑦 = ( 𝑧 cyclShift 𝑛 ) ↔ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ) |
20 |
19
|
cbvrexvw |
⊢ ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ↔ ∃ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) |
21 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
22 |
21
|
clwwlkbp |
⊢ ( 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑧 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑧 ≠ ∅ ) ) |
23 |
22
|
simp2d |
⊢ ( 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) → 𝑧 ∈ Word ( Vtx ‘ 𝐺 ) ) |
24 |
23
|
ad2antlr |
⊢ ( ( ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → 𝑧 ∈ Word ( Vtx ‘ 𝐺 ) ) |
25 |
|
simpr |
⊢ ( ( ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) |
26 |
24 25
|
cshwcsh2id |
⊢ ( ( ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ∧ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ∧ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
27 |
26
|
exp5l |
⊢ ( ( ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) → ( 𝑥 = ( 𝑦 cyclShift 𝑚 ) → ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) → ( 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) ) ) |
28 |
27
|
imp41 |
⊢ ( ( ( ( ( ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ∧ 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
29 |
28
|
rexlimdva |
⊢ ( ( ( ( ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ∧ 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) → ( ∃ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
30 |
29
|
rexlimdva2 |
⊢ ( ( ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( ∃ 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑚 ) → ( ∃ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
31 |
20 30
|
syl7bi |
⊢ ( ( ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( ∃ 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑚 ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
32 |
17 31
|
syl5bi |
⊢ ( ( ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
33 |
32
|
exp31 |
⊢ ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) → ( 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) → ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) ) ) |
34 |
33
|
com15 |
⊢ ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) → ( 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) → ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) ) ) |
35 |
34
|
impcom |
⊢ ( ( 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) ) |
36 |
35
|
3adant1 |
⊢ ( ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) ) |
37 |
36
|
impcom |
⊢ ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
38 |
37
|
com13 |
⊢ ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
39 |
38
|
3impia |
⊢ ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) → ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
40 |
39
|
impcom |
⊢ ( ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ∧ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) |
41 |
13 14 40
|
3jca |
⊢ ( ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ∧ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) → ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
42 |
1
|
erclwwlkeq |
⊢ ( ( 𝑥 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑧 ↔ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
43 |
42
|
3adant2 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑧 ↔ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
44 |
41 43
|
syl5ibrcom |
⊢ ( ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ∧ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) → ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → 𝑥 ∼ 𝑧 ) ) |
45 |
44
|
exp31 |
⊢ ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) → ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → 𝑥 ∼ 𝑧 ) ) ) ) |
46 |
45
|
com24 |
⊢ ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) → ( ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → 𝑥 ∼ 𝑧 ) ) ) ) |
47 |
46
|
ex |
⊢ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) → ( ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → 𝑥 ∼ 𝑧 ) ) ) ) ) |
48 |
47
|
com4t |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → 𝑥 ∼ 𝑧 ) ) ) ) ) |
49 |
12 48
|
sylbid |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑦 → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → 𝑥 ∼ 𝑧 ) ) ) ) ) |
50 |
49
|
com25 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( ( 𝑦 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑧 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( 𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧 ) ) ) ) ) |
51 |
10 50
|
sylbid |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( 𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧 ) ) ) ) ) |
52 |
8 51
|
mpdd |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( 𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧 ) ) ) ) |
53 |
52
|
com24 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑦 → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( 𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧 ) ) ) ) |
54 |
6 53
|
mpdd |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑦 → ( 𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧 ) ) ) |
55 |
54
|
impd |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( ( 𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧 ) → 𝑥 ∼ 𝑧 ) ) |
56 |
2 3 4 55
|
mp3an |
⊢ ( ( 𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧 ) → 𝑥 ∼ 𝑧 ) |