Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
2 |
1
|
clwwlkbp |
⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ) |
3 |
|
cshw0 |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( 𝑊 cyclShift 0 ) = 𝑊 ) |
4 |
3
|
3ad2ant2 |
⊢ ( ( 𝐺 ∈ V ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) → ( 𝑊 cyclShift 0 ) = 𝑊 ) |
5 |
4
|
eleq1d |
⊢ ( ( 𝐺 ∈ V ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) → ( ( 𝑊 cyclShift 0 ) ∈ ( ClWWalks ‘ 𝐺 ) ↔ 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
6 |
5
|
biimprd |
⊢ ( ( 𝐺 ∈ V ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) → ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝑊 cyclShift 0 ) ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
7 |
2 6
|
mpcom |
⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝑊 cyclShift 0 ) ∈ ( ClWWalks ‘ 𝐺 ) ) |
8 |
|
oveq2 |
⊢ ( 𝑁 = 0 → ( 𝑊 cyclShift 𝑁 ) = ( 𝑊 cyclShift 0 ) ) |
9 |
8
|
eleq1d |
⊢ ( 𝑁 = 0 → ( ( 𝑊 cyclShift 𝑁 ) ∈ ( ClWWalks ‘ 𝐺 ) ↔ ( 𝑊 cyclShift 0 ) ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
10 |
7 9
|
syl5ibrcom |
⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝑁 = 0 → ( 𝑊 cyclShift 𝑁 ) ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
11 |
10
|
adantr |
⊢ ( ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑁 = 0 → ( 𝑊 cyclShift 𝑁 ) ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
12 |
|
fzo1fzo0n0 |
⊢ ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ↔ ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ∧ 𝑁 ≠ 0 ) ) |
13 |
|
cshwcl |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( 𝑊 cyclShift 𝑁 ) ∈ Word ( Vtx ‘ 𝐺 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) → ( 𝑊 cyclShift 𝑁 ) ∈ Word ( Vtx ‘ 𝐺 ) ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) → ( 𝑊 cyclShift 𝑁 ) ∈ Word ( Vtx ‘ 𝐺 ) ) |
16 |
15
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 cyclShift 𝑁 ) ∈ Word ( Vtx ‘ 𝐺 ) ) |
17 |
|
simpl |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) → 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) |
18 |
|
elfzoelz |
⊢ ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → 𝑁 ∈ ℤ ) |
19 |
|
cshwlen |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ℤ ) → ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) = ( ♯ ‘ 𝑊 ) ) |
20 |
17 18 19
|
syl2an |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) = ( ♯ ‘ 𝑊 ) ) |
21 |
|
hasheq0 |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( ( ♯ ‘ 𝑊 ) = 0 ↔ 𝑊 = ∅ ) ) |
22 |
21
|
bicomd |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( 𝑊 = ∅ ↔ ( ♯ ‘ 𝑊 ) = 0 ) ) |
23 |
22
|
necon3bid |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( 𝑊 ≠ ∅ ↔ ( ♯ ‘ 𝑊 ) ≠ 0 ) ) |
24 |
23
|
biimpa |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) → ( ♯ ‘ 𝑊 ) ≠ 0 ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ 𝑊 ) ≠ 0 ) |
26 |
20 25
|
eqnetrd |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) ≠ 0 ) |
27 |
14
|
adantr |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 cyclShift 𝑁 ) ∈ Word ( Vtx ‘ 𝐺 ) ) |
28 |
|
hasheq0 |
⊢ ( ( 𝑊 cyclShift 𝑁 ) ∈ Word ( Vtx ‘ 𝐺 ) → ( ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) = 0 ↔ ( 𝑊 cyclShift 𝑁 ) = ∅ ) ) |
29 |
27 28
|
syl |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) = 0 ↔ ( 𝑊 cyclShift 𝑁 ) = ∅ ) ) |
30 |
29
|
necon3bid |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) ≠ 0 ↔ ( 𝑊 cyclShift 𝑁 ) ≠ ∅ ) ) |
31 |
26 30
|
mpbid |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 cyclShift 𝑁 ) ≠ ∅ ) |
32 |
31
|
3ad2antl1 |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 cyclShift 𝑁 ) ≠ ∅ ) |
33 |
16 32
|
jca |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 cyclShift 𝑁 ) ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( 𝑊 cyclShift 𝑁 ) ≠ ∅ ) ) |
34 |
17
|
3ad2ant1 |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) → 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) |
35 |
34
|
anim1i |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ) |
36 |
|
3simpc |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
37 |
36
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
38 |
|
clwwisshclwwslem |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) → ∀ 𝑗 ∈ ( 0 ..^ ( ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) − 1 ) ) { ( ( 𝑊 cyclShift 𝑁 ) ‘ 𝑗 ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ ( 𝑗 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
39 |
35 37 38
|
sylc |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑗 ∈ ( 0 ..^ ( ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) − 1 ) ) { ( ( 𝑊 cyclShift 𝑁 ) ‘ 𝑗 ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ ( 𝑗 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) |
40 |
|
elfzofz |
⊢ ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) |
41 |
|
lswcshw |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) = ( 𝑊 ‘ ( 𝑁 − 1 ) ) ) |
42 |
40 41
|
sylan2 |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) = ( 𝑊 ‘ ( 𝑁 − 1 ) ) ) |
43 |
|
fzo0ss1 |
⊢ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) |
44 |
43
|
sseli |
⊢ ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
45 |
|
cshwidx0 |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) = ( 𝑊 ‘ 𝑁 ) ) |
46 |
44 45
|
sylan2 |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) = ( 𝑊 ‘ 𝑁 ) ) |
47 |
42 46
|
preq12d |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → { ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) } = { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ) |
48 |
47
|
ex |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → { ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) } = { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ) ) |
49 |
48
|
adantr |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) → ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → { ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) } = { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ) ) |
50 |
49
|
3ad2ant1 |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) → ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → { ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) } = { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ) ) |
51 |
50
|
imp |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → { ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) } = { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ) |
52 |
|
elfzo1elm1fzo0 |
⊢ ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → ( 𝑁 − 1 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
53 |
52
|
adantl |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑁 − 1 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
54 |
|
fveq2 |
⊢ ( 𝑖 = ( 𝑁 − 1 ) → ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ ( 𝑁 − 1 ) ) ) |
55 |
54
|
adantl |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑖 = ( 𝑁 − 1 ) ) → ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ ( 𝑁 − 1 ) ) ) |
56 |
|
fvoveq1 |
⊢ ( 𝑖 = ( 𝑁 − 1 ) → ( 𝑊 ‘ ( 𝑖 + 1 ) ) = ( 𝑊 ‘ ( ( 𝑁 − 1 ) + 1 ) ) ) |
57 |
18
|
zcnd |
⊢ ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → 𝑁 ∈ ℂ ) |
58 |
57
|
adantl |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝑁 ∈ ℂ ) |
59 |
|
1cnd |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 1 ∈ ℂ ) |
60 |
58 59
|
npcand |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 ) |
61 |
60
|
fveq2d |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ ( ( 𝑁 − 1 ) + 1 ) ) = ( 𝑊 ‘ 𝑁 ) ) |
62 |
56 61
|
sylan9eqr |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑖 = ( 𝑁 − 1 ) ) → ( 𝑊 ‘ ( 𝑖 + 1 ) ) = ( 𝑊 ‘ 𝑁 ) ) |
63 |
55 62
|
preq12d |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑖 = ( 𝑁 − 1 ) ) → { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } = { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ) |
64 |
63
|
eleq1d |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑖 = ( 𝑁 − 1 ) ) → ( { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
65 |
53 64
|
rspcdv |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
66 |
65
|
a1d |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
67 |
66
|
ex |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) ) |
68 |
67
|
adantr |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) → ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) ) |
69 |
68
|
com24 |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) → ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) ) |
70 |
69
|
3imp1 |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 𝑁 ) } ∈ ( Edg ‘ 𝐺 ) ) |
71 |
51 70
|
eqeltrd |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → { ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) |
72 |
33 39 71
|
3jca |
⊢ ( ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝑊 cyclShift 𝑁 ) ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( 𝑊 cyclShift 𝑁 ) ≠ ∅ ) ∧ ∀ 𝑗 ∈ ( 0 ..^ ( ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) − 1 ) ) { ( ( 𝑊 cyclShift 𝑁 ) ‘ 𝑗 ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ ( 𝑗 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
73 |
72
|
expcom |
⊢ ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) → ( ( ( 𝑊 cyclShift 𝑁 ) ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( 𝑊 cyclShift 𝑁 ) ≠ ∅ ) ∧ ∀ 𝑗 ∈ ( 0 ..^ ( ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) − 1 ) ) { ( ( 𝑊 cyclShift 𝑁 ) ‘ 𝑗 ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ ( 𝑗 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
74 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
75 |
1 74
|
isclwwlk |
⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ↔ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
76 |
1 74
|
isclwwlk |
⊢ ( ( 𝑊 cyclShift 𝑁 ) ∈ ( ClWWalks ‘ 𝐺 ) ↔ ( ( ( 𝑊 cyclShift 𝑁 ) ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( 𝑊 cyclShift 𝑁 ) ≠ ∅ ) ∧ ∀ 𝑗 ∈ ( 0 ..^ ( ( ♯ ‘ ( 𝑊 cyclShift 𝑁 ) ) − 1 ) ) { ( ( 𝑊 cyclShift 𝑁 ) ‘ 𝑗 ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ ( 𝑗 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ ( 𝑊 cyclShift 𝑁 ) ) , ( ( 𝑊 cyclShift 𝑁 ) ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
77 |
73 75 76
|
3imtr4g |
⊢ ( 𝑁 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝑊 cyclShift 𝑁 ) ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
78 |
12 77
|
sylbir |
⊢ ( ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ∧ 𝑁 ≠ 0 ) → ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝑊 cyclShift 𝑁 ) ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
79 |
78
|
expcom |
⊢ ( 𝑁 ≠ 0 → ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝑊 cyclShift 𝑁 ) ∈ ( ClWWalks ‘ 𝐺 ) ) ) ) |
80 |
79
|
com13 |
⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → ( 𝑁 ≠ 0 → ( 𝑊 cyclShift 𝑁 ) ∈ ( ClWWalks ‘ 𝐺 ) ) ) ) |
81 |
80
|
imp |
⊢ ( ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑁 ≠ 0 → ( 𝑊 cyclShift 𝑁 ) ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
82 |
11 81
|
pm2.61dne |
⊢ ( ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 cyclShift 𝑁 ) ∈ ( ClWWalks ‘ 𝐺 ) ) |