| 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 |
|
vex |
⊢ 𝑥 ∈ V |
| 4 |
|
vex |
⊢ 𝑦 ∈ V |
| 5 |
|
vex |
⊢ 𝑧 ∈ V |
| 6 |
1 2
|
erclwwlkneqlen |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) → ( 𝑥 ∼ 𝑦 → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) |
| 7 |
6
|
3adant3 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑦 → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) |
| 8 |
1 2
|
erclwwlkneqlen |
⊢ ( ( 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 → ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ) ) |
| 9 |
8
|
3adant1 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 → ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ) ) |
| 10 |
1 2
|
erclwwlkneq |
⊢ ( ( 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 ↔ ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
| 11 |
10
|
3adant1 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 ↔ ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
| 12 |
1 2
|
erclwwlkneq |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) → ( 𝑥 ∼ 𝑦 ↔ ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) ) |
| 13 |
12
|
3adant3 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑦 ↔ ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) ) |
| 14 |
|
simpr1 |
⊢ ( ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ∧ ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) → 𝑥 ∈ 𝑊 ) |
| 15 |
|
simplr2 |
⊢ ( ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ∧ ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) → 𝑧 ∈ 𝑊 ) |
| 16 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝑦 cyclShift 𝑛 ) = ( 𝑦 cyclShift 𝑚 ) ) |
| 17 |
16
|
eqeq2d |
⊢ ( 𝑛 = 𝑚 → ( 𝑥 = ( 𝑦 cyclShift 𝑛 ) ↔ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ) |
| 18 |
17
|
cbvrexvw |
⊢ ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ↔ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) |
| 19 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝑧 cyclShift 𝑛 ) = ( 𝑧 cyclShift 𝑘 ) ) |
| 20 |
19
|
eqeq2d |
⊢ ( 𝑛 = 𝑘 → ( 𝑦 = ( 𝑧 cyclShift 𝑛 ) ↔ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ) |
| 21 |
20
|
cbvrexvw |
⊢ ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ↔ ∃ 𝑘 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) |
| 22 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
| 23 |
22
|
clwwlknbp |
⊢ ( 𝑧 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( 𝑧 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑧 ) = 𝑁 ) ) |
| 24 |
|
eqcom |
⊢ ( ( ♯ ‘ 𝑧 ) = 𝑁 ↔ 𝑁 = ( ♯ ‘ 𝑧 ) ) |
| 25 |
24
|
biimpi |
⊢ ( ( ♯ ‘ 𝑧 ) = 𝑁 → 𝑁 = ( ♯ ‘ 𝑧 ) ) |
| 26 |
23 25
|
simpl2im |
⊢ ( 𝑧 ∈ ( 𝑁 ClWWalksN 𝐺 ) → 𝑁 = ( ♯ ‘ 𝑧 ) ) |
| 27 |
26 1
|
eleq2s |
⊢ ( 𝑧 ∈ 𝑊 → 𝑁 = ( ♯ ‘ 𝑧 ) ) |
| 28 |
27
|
ad2antlr |
⊢ ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → 𝑁 = ( ♯ ‘ 𝑧 ) ) |
| 29 |
23
|
simpld |
⊢ ( 𝑧 ∈ ( 𝑁 ClWWalksN 𝐺 ) → 𝑧 ∈ Word ( Vtx ‘ 𝐺 ) ) |
| 30 |
29 1
|
eleq2s |
⊢ ( 𝑧 ∈ 𝑊 → 𝑧 ∈ Word ( Vtx ‘ 𝐺 ) ) |
| 31 |
30
|
ad2antlr |
⊢ ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → 𝑧 ∈ Word ( Vtx ‘ 𝐺 ) ) |
| 32 |
31
|
adantl |
⊢ ( ( 𝑁 = ( ♯ ‘ 𝑧 ) ∧ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ) → 𝑧 ∈ Word ( Vtx ‘ 𝐺 ) ) |
| 33 |
|
simprr |
⊢ ( ( 𝑁 = ( ♯ ‘ 𝑧 ) ∧ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ) → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) |
| 34 |
32 33
|
cshwcsh2id |
⊢ ( ( 𝑁 = ( ♯ ‘ 𝑧 ) ∧ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 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 ) ) ) |
| 35 |
|
oveq2 |
⊢ ( 𝑁 = ( ♯ ‘ 𝑧 ) → ( 0 ... 𝑁 ) = ( 0 ... ( ♯ ‘ 𝑧 ) ) ) |
| 36 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝑧 ) = ( ♯ ‘ 𝑦 ) → ( 0 ... ( ♯ ‘ 𝑧 ) ) = ( 0 ... ( ♯ ‘ 𝑦 ) ) ) |
| 37 |
36
|
eqcoms |
⊢ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( 0 ... ( ♯ ‘ 𝑧 ) ) = ( 0 ... ( ♯ ‘ 𝑦 ) ) ) |
| 38 |
37
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( 0 ... ( ♯ ‘ 𝑧 ) ) = ( 0 ... ( ♯ ‘ 𝑦 ) ) ) |
| 39 |
38
|
adantl |
⊢ ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( 0 ... ( ♯ ‘ 𝑧 ) ) = ( 0 ... ( ♯ ‘ 𝑦 ) ) ) |
| 40 |
35 39
|
sylan9eq |
⊢ ( ( 𝑁 = ( ♯ ‘ 𝑧 ) ∧ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ) → ( 0 ... 𝑁 ) = ( 0 ... ( ♯ ‘ 𝑦 ) ) ) |
| 41 |
40
|
eleq2d |
⊢ ( ( 𝑁 = ( ♯ ‘ 𝑧 ) ∧ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ) → ( 𝑚 ∈ ( 0 ... 𝑁 ) ↔ 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ) ) |
| 42 |
41
|
anbi1d |
⊢ ( ( 𝑁 = ( ♯ ‘ 𝑧 ) ∧ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ) → ( ( 𝑚 ∈ ( 0 ... 𝑁 ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ↔ ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ) ) |
| 43 |
35
|
eleq2d |
⊢ ( 𝑁 = ( ♯ ‘ 𝑧 ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) ↔ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ) |
| 44 |
43
|
anbi1d |
⊢ ( 𝑁 = ( ♯ ‘ 𝑧 ) → ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ↔ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ∧ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ) ) |
| 45 |
44
|
adantr |
⊢ ( ( 𝑁 = ( ♯ ‘ 𝑧 ) ∧ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ) → ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ↔ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ∧ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ) ) |
| 46 |
42 45
|
anbi12d |
⊢ ( ( 𝑁 = ( ♯ ‘ 𝑧 ) ∧ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ) → ( ( ( 𝑚 ∈ ( 0 ... 𝑁 ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ) ↔ ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ∧ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ∧ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ) ) ) |
| 47 |
35
|
rexeqdv |
⊢ ( 𝑁 = ( ♯ ‘ 𝑧 ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ↔ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
| 48 |
47
|
adantr |
⊢ ( ( 𝑁 = ( ♯ ‘ 𝑧 ) ∧ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ↔ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
| 49 |
34 46 48
|
3imtr4d |
⊢ ( ( 𝑁 = ( ♯ ‘ 𝑧 ) ∧ ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ) → ( ( ( 𝑚 ∈ ( 0 ... 𝑁 ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
| 50 |
28 49
|
mpancom |
⊢ ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( ( ( 𝑚 ∈ ( 0 ... 𝑁 ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
| 51 |
50
|
exp5l |
⊢ ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( 𝑚 ∈ ( 0 ... 𝑁 ) → ( 𝑥 = ( 𝑦 cyclShift 𝑚 ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) ) ) |
| 52 |
51
|
imp41 |
⊢ ( ( ( ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ∧ 𝑚 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → ( 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
| 53 |
52
|
rexlimdva |
⊢ ( ( ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ∧ 𝑚 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) → ( ∃ 𝑘 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
| 54 |
53
|
ex |
⊢ ( ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) ∧ 𝑚 ∈ ( 0 ... 𝑁 ) ) → ( 𝑥 = ( 𝑦 cyclShift 𝑚 ) → ( ∃ 𝑘 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
| 55 |
54
|
rexlimdva |
⊢ ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑚 ) → ( ∃ 𝑘 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
| 56 |
21 55
|
syl7bi |
⊢ ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑚 ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
| 57 |
18 56
|
biimtrid |
⊢ ( ( ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ∧ 𝑧 ∈ 𝑊 ) ∧ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
| 58 |
57
|
exp31 |
⊢ ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) → ( 𝑧 ∈ 𝑊 → ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) ) ) |
| 59 |
58
|
com15 |
⊢ ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) → ( 𝑧 ∈ 𝑊 → ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) ) ) |
| 60 |
59
|
impcom |
⊢ ( ( 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) ) |
| 61 |
60
|
3adant1 |
⊢ ( ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) ) |
| 62 |
61
|
impcom |
⊢ ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
| 63 |
62
|
com13 |
⊢ ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) → ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
| 64 |
63
|
3impia |
⊢ ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) → ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
| 65 |
64
|
impcom |
⊢ ( ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ∧ ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) |
| 66 |
14 15 65
|
3jca |
⊢ ( ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ∧ ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) → ( 𝑥 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |
| 67 |
1 2
|
erclwwlkneq |
⊢ ( ( 𝑥 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑧 ↔ ( 𝑥 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
| 68 |
67
|
3adant2 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑧 ↔ ( 𝑥 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) ) |
| 69 |
66 68
|
syl5ibrcom |
⊢ ( ( ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ∧ ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) ) ∧ ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) ) → ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → 𝑥 ∼ 𝑧 ) ) |
| 70 |
69
|
exp31 |
⊢ ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) → ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → 𝑥 ∼ 𝑧 ) ) ) ) |
| 71 |
70
|
com24 |
⊢ ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) → ( ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → 𝑥 ∼ 𝑧 ) ) ) ) |
| 72 |
71
|
ex |
⊢ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) → ( ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → 𝑥 ∼ 𝑧 ) ) ) ) ) |
| 73 |
72
|
com4t |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑥 = ( 𝑦 cyclShift 𝑛 ) ) → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → 𝑥 ∼ 𝑧 ) ) ) ) ) |
| 74 |
13 73
|
sylbid |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑦 → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → 𝑥 ∼ 𝑧 ) ) ) ) ) |
| 75 |
74
|
com25 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( ( 𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑦 = ( 𝑧 cyclShift 𝑛 ) ) → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( 𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧 ) ) ) ) ) |
| 76 |
11 75
|
sylbid |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( 𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧 ) ) ) ) ) |
| 77 |
9 76
|
mpdd |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑦 ∼ 𝑧 → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( 𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧 ) ) ) ) |
| 78 |
77
|
com24 |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑦 → ( ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) → ( 𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧 ) ) ) ) |
| 79 |
7 78
|
mpdd |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( 𝑥 ∼ 𝑦 → ( 𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧 ) ) ) |
| 80 |
79
|
impd |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → ( ( 𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧 ) → 𝑥 ∼ 𝑧 ) ) |
| 81 |
3 4 5 80
|
mp3an |
⊢ ( ( 𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧 ) → 𝑥 ∼ 𝑧 ) |