Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlkn1.w |
⊢ 𝑊 = ( 𝑁 ClWWalksN 𝐺 ) |
2 |
|
simpl |
⊢ ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) → 𝑘 ∈ ( 0 ... 𝑁 ) ) |
3 |
2
|
anim1i |
⊢ ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) ) |
4 |
3
|
adantr |
⊢ ( ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑥 cyclShift 𝑚 ) ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) ) |
5 |
|
simpr |
⊢ ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) → 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) |
6 |
5
|
adantr |
⊢ ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) |
7 |
6
|
anim1i |
⊢ ( ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑥 cyclShift 𝑚 ) ) → ( 𝑋 = ( 𝑥 cyclShift 𝑘 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑥 cyclShift 𝑚 ) ) ) |
8 |
1
|
eleclclwwlknlem1 |
⊢ ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → ( ( 𝑋 = ( 𝑥 cyclShift 𝑘 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑥 cyclShift 𝑚 ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑋 cyclShift 𝑛 ) ) ) |
9 |
4 7 8
|
sylc |
⊢ ( ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑥 cyclShift 𝑚 ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑋 cyclShift 𝑛 ) ) |
10 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
11 |
10
|
clwwlknbp |
⊢ ( 𝑥 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑥 ) = 𝑁 ) ) |
12 |
11 1
|
eleq2s |
⊢ ( 𝑥 ∈ 𝑊 → ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑥 ) = 𝑁 ) ) |
13 |
|
fznn0sub2 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( 𝑁 − 𝑘 ) ∈ ( 0 ... 𝑁 ) ) |
14 |
|
oveq1 |
⊢ ( ( ♯ ‘ 𝑥 ) = 𝑁 → ( ( ♯ ‘ 𝑥 ) − 𝑘 ) = ( 𝑁 − 𝑘 ) ) |
15 |
14
|
eleq1d |
⊢ ( ( ♯ ‘ 𝑥 ) = 𝑁 → ( ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ↔ ( 𝑁 − 𝑘 ) ∈ ( 0 ... 𝑁 ) ) ) |
16 |
13 15
|
syl5ibr |
⊢ ( ( ♯ ‘ 𝑥 ) = 𝑁 → ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑥 ) = 𝑁 ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ) ) |
18 |
12 17
|
syl |
⊢ ( 𝑥 ∈ 𝑊 → ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ) ) |
20 |
19
|
com12 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) → ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ) ) |
21 |
20
|
adantr |
⊢ ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) → ( ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) → ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ) ) |
22 |
21
|
imp |
⊢ ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ) |
23 |
22
|
adantr |
⊢ ( ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑋 cyclShift 𝑛 ) ) → ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ) |
24 |
|
simpr |
⊢ ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) |
25 |
24
|
ancomd |
⊢ ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → ( 𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊 ) ) |
26 |
25
|
adantr |
⊢ ( ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑋 cyclShift 𝑛 ) ) → ( 𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊 ) ) |
27 |
23 26
|
jca |
⊢ ( ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑋 cyclShift 𝑛 ) ) → ( ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ∧ ( 𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊 ) ) ) |
28 |
|
simpll |
⊢ ( ( ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑥 ) = 𝑁 ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ) |
29 |
|
oveq2 |
⊢ ( 𝑁 = ( ♯ ‘ 𝑥 ) → ( 0 ... 𝑁 ) = ( 0 ... ( ♯ ‘ 𝑥 ) ) ) |
30 |
29
|
eleq2d |
⊢ ( 𝑁 = ( ♯ ‘ 𝑥 ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) ↔ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ) |
31 |
30
|
eqcoms |
⊢ ( ( ♯ ‘ 𝑥 ) = 𝑁 → ( 𝑘 ∈ ( 0 ... 𝑁 ) ↔ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ) |
32 |
31
|
adantl |
⊢ ( ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑥 ) = 𝑁 ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) ↔ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ) |
33 |
32
|
biimpa |
⊢ ( ( ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑥 ) = 𝑁 ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) |
34 |
28 33
|
jca |
⊢ ( ( ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑥 ) = 𝑁 ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ) |
35 |
34
|
ex |
⊢ ( ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑥 ) = 𝑁 ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ) ) |
36 |
12 35
|
syl |
⊢ ( 𝑥 ∈ 𝑊 → ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ) ) |
37 |
36
|
adantl |
⊢ ( ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ) ) |
38 |
37
|
com12 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑁 ) → ( ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) → ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ) ) |
39 |
38
|
adantr |
⊢ ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) → ( ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) → ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ) ) |
40 |
39
|
imp |
⊢ ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ) |
41 |
5
|
eqcomd |
⊢ ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) → ( 𝑥 cyclShift 𝑘 ) = 𝑋 ) |
42 |
41
|
adantr |
⊢ ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → ( 𝑥 cyclShift 𝑘 ) = 𝑋 ) |
43 |
|
oveq1 |
⊢ ( 𝑋 = ( 𝑥 cyclShift 𝑘 ) → ( 𝑋 cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) = ( ( 𝑥 cyclShift 𝑘 ) cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) ) |
44 |
43
|
eqcoms |
⊢ ( ( 𝑥 cyclShift 𝑘 ) = 𝑋 → ( 𝑋 cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) = ( ( 𝑥 cyclShift 𝑘 ) cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) ) |
45 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) → 𝑘 ∈ ℤ ) |
46 |
|
2cshwid |
⊢ ( ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ℤ ) → ( ( 𝑥 cyclShift 𝑘 ) cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) = 𝑥 ) |
47 |
45 46
|
sylan2 |
⊢ ( ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) → ( ( 𝑥 cyclShift 𝑘 ) cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) = 𝑥 ) |
48 |
44 47
|
sylan9eqr |
⊢ ( ( ( 𝑥 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) ∧ ( 𝑥 cyclShift 𝑘 ) = 𝑋 ) → ( 𝑋 cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) = 𝑥 ) |
49 |
40 42 48
|
syl2anc |
⊢ ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → ( 𝑋 cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) = 𝑥 ) |
50 |
49
|
eqcomd |
⊢ ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → 𝑥 = ( 𝑋 cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) ) |
51 |
50
|
anim1i |
⊢ ( ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑋 cyclShift 𝑛 ) ) → ( 𝑥 = ( 𝑋 cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑋 cyclShift 𝑛 ) ) ) |
52 |
1
|
eleclclwwlknlem1 |
⊢ ( ( ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ∈ ( 0 ... 𝑁 ) ∧ ( 𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊 ) ) → ( ( 𝑥 = ( 𝑋 cyclShift ( ( ♯ ‘ 𝑥 ) − 𝑘 ) ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑋 cyclShift 𝑛 ) ) → ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑥 cyclShift 𝑚 ) ) ) |
53 |
27 51 52
|
sylc |
⊢ ( ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑋 cyclShift 𝑛 ) ) → ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑥 cyclShift 𝑚 ) ) |
54 |
9 53
|
impbida |
⊢ ( ( ( 𝑘 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑥 cyclShift 𝑘 ) ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ) ) → ( ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑥 cyclShift 𝑚 ) ↔ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑌 = ( 𝑋 cyclShift 𝑛 ) ) ) |