| Step |
Hyp |
Ref |
Expression |
| 1 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
| 2 |
1
|
wlkf |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) |
| 3 |
|
revcl |
⊢ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) → ( reverse ‘ 𝐹 ) ∈ Word dom ( iEdg ‘ 𝐺 ) ) |
| 4 |
2 3
|
syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( reverse ‘ 𝐹 ) ∈ Word dom ( iEdg ‘ 𝐺 ) ) |
| 5 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
| 6 |
5
|
wlkpwrd |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) ) |
| 7 |
|
revcl |
⊢ ( 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) → ( reverse ‘ 𝑃 ) ∈ Word ( Vtx ‘ 𝐺 ) ) |
| 8 |
|
wrdf |
⊢ ( ( reverse ‘ 𝑃 ) ∈ Word ( Vtx ‘ 𝐺 ) → ( reverse ‘ 𝑃 ) : ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝑃 ) ) ) ⟶ ( Vtx ‘ 𝐺 ) ) |
| 9 |
6 7 8
|
3syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( reverse ‘ 𝑃 ) : ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝑃 ) ) ) ⟶ ( Vtx ‘ 𝐺 ) ) |
| 10 |
|
revlen |
⊢ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) → ( ♯ ‘ ( reverse ‘ 𝐹 ) ) = ( ♯ ‘ 𝐹 ) ) |
| 11 |
2 10
|
syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ ( reverse ‘ 𝐹 ) ) = ( ♯ ‘ 𝐹 ) ) |
| 12 |
11
|
oveq2d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 0 ... ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) = ( 0 ... ( ♯ ‘ 𝐹 ) ) ) |
| 13 |
|
wlklenvp1 |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) |
| 14 |
13
|
oveq2d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 0 ..^ ( ♯ ‘ 𝑃 ) ) = ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ) |
| 15 |
|
revlen |
⊢ ( 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) → ( ♯ ‘ ( reverse ‘ 𝑃 ) ) = ( ♯ ‘ 𝑃 ) ) |
| 16 |
6 15
|
syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ ( reverse ‘ 𝑃 ) ) = ( ♯ ‘ 𝑃 ) ) |
| 17 |
16
|
oveq2d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝑃 ) ) ) = ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ) |
| 18 |
|
wlkcl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
| 19 |
18
|
nn0zd |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℤ ) |
| 20 |
|
fzval3 |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℤ → ( 0 ... ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ) |
| 21 |
19 20
|
syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 0 ... ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ) |
| 22 |
14 17 21
|
3eqtr4rd |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 0 ... ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝑃 ) ) ) ) |
| 23 |
12 22
|
eqtrd |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 0 ... ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) = ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝑃 ) ) ) ) |
| 24 |
23
|
feq2d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( reverse ‘ 𝑃 ) : ( 0 ... ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) ⟶ ( Vtx ‘ 𝐺 ) ↔ ( reverse ‘ 𝑃 ) : ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝑃 ) ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) |
| 25 |
9 24
|
mpbird |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( reverse ‘ 𝑃 ) : ( 0 ... ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) ⟶ ( Vtx ‘ 𝐺 ) ) |
| 26 |
11
|
oveq2d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) = ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 27 |
26
|
eleq2d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝑘 ∈ ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) ↔ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
| 28 |
27
|
biimpa |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) ) → 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 29 |
|
revfv |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) = ( 𝐹 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) − 𝑘 ) ) ) |
| 30 |
2 29
|
sylan |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) = ( 𝐹 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) − 𝑘 ) ) ) |
| 31 |
|
wlklenvm1 |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) ) |
| 32 |
31
|
oveq1d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( ♯ ‘ 𝐹 ) − 1 ) = ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 1 ) ) |
| 33 |
|
lencl |
⊢ ( 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) → ( ♯ ‘ 𝑃 ) ∈ ℕ0 ) |
| 34 |
33
|
nn0cnd |
⊢ ( 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) → ( ♯ ‘ 𝑃 ) ∈ ℂ ) |
| 35 |
|
sub1m1 |
⊢ ( ( ♯ ‘ 𝑃 ) ∈ ℂ → ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 1 ) = ( ( ♯ ‘ 𝑃 ) − 2 ) ) |
| 36 |
6 34 35
|
3syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 1 ) = ( ( ♯ ‘ 𝑃 ) − 2 ) ) |
| 37 |
32 36
|
eqtrd |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( ♯ ‘ 𝐹 ) − 1 ) = ( ( ♯ ‘ 𝑃 ) − 2 ) ) |
| 38 |
37
|
fvoveq1d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐹 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) − 𝑘 ) ) = ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) |
| 39 |
38
|
adantr |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) − 𝑘 ) ) = ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) |
| 40 |
30 39
|
eqtrd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) = ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) |
| 41 |
40
|
fveq2d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) |
| 42 |
41
|
adantr |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) |
| 43 |
|
fzonn0p1p1 |
⊢ ( 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝑘 + 1 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ) |
| 44 |
43
|
adantl |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑘 + 1 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ) |
| 45 |
14
|
adantr |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 0 ..^ ( ♯ ‘ 𝑃 ) ) = ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ) |
| 46 |
44 45
|
eleqtrrd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑘 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ) |
| 47 |
|
revfv |
⊢ ( ( 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( 𝑘 + 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ) → ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 1 ) − ( 𝑘 + 1 ) ) ) ) |
| 48 |
6 46 47
|
syl2an2r |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 1 ) − ( 𝑘 + 1 ) ) ) ) |
| 49 |
|
elfzoelz |
⊢ ( 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → 𝑘 ∈ ℤ ) |
| 50 |
49
|
zcnd |
⊢ ( 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → 𝑘 ∈ ℂ ) |
| 51 |
50
|
adantl |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑘 ∈ ℂ ) |
| 52 |
|
1cnd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 1 ∈ ℂ ) |
| 53 |
51 52
|
addcomd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑘 + 1 ) = ( 1 + 𝑘 ) ) |
| 54 |
53
|
oveq2d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝑃 ) − 1 ) − ( 𝑘 + 1 ) ) = ( ( ( ♯ ‘ 𝑃 ) − 1 ) − ( 1 + 𝑘 ) ) ) |
| 55 |
6 34
|
syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝑃 ) ∈ ℂ ) |
| 56 |
55
|
adantr |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ♯ ‘ 𝑃 ) ∈ ℂ ) |
| 57 |
56 52
|
subcld |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ♯ ‘ 𝑃 ) − 1 ) ∈ ℂ ) |
| 58 |
57 52 51
|
subsub4d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 1 ) − 𝑘 ) = ( ( ( ♯ ‘ 𝑃 ) − 1 ) − ( 1 + 𝑘 ) ) ) |
| 59 |
36
|
oveq1d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 1 ) − 𝑘 ) = ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) |
| 60 |
59
|
adantr |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 1 ) − 𝑘 ) = ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) |
| 61 |
54 58 60
|
3eqtr2d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝑃 ) − 1 ) − ( 𝑘 + 1 ) ) = ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) |
| 62 |
61
|
fveq2d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 1 ) − ( 𝑘 + 1 ) ) ) = ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) |
| 63 |
48 62
|
eqtrd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) |
| 64 |
63
|
sneqd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → { ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) } ) |
| 65 |
64
|
adantr |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ) → { ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) } ) |
| 66 |
|
sneq |
⊢ ( ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) → { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } = { ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } ) |
| 67 |
66
|
adantl |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ) → { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } = { ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } ) |
| 68 |
|
eqcom |
⊢ ( ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ↔ ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) = ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) ) |
| 69 |
|
fzossfzop1 |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ) |
| 70 |
18 69
|
syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ) |
| 71 |
70 14
|
sseqtrrd |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ) |
| 72 |
71
|
sselda |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ) |
| 73 |
|
revfv |
⊢ ( ( 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ) → ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 𝑘 ) ) ) |
| 74 |
6 72 73
|
syl2an2r |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 𝑘 ) ) ) |
| 75 |
57 51 52
|
sub32d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 𝑘 ) − 1 ) = ( ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 1 ) − 𝑘 ) ) |
| 76 |
75
|
oveq1d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 𝑘 ) − 1 ) + 1 ) = ( ( ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 1 ) − 𝑘 ) + 1 ) ) |
| 77 |
57 51
|
subcld |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 𝑘 ) ∈ ℂ ) |
| 78 |
77 52
|
npcand |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 𝑘 ) − 1 ) + 1 ) = ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 𝑘 ) ) |
| 79 |
59
|
oveq1d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 1 ) − 𝑘 ) + 1 ) = ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) |
| 80 |
79
|
adantr |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 1 ) − 𝑘 ) + 1 ) = ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) |
| 81 |
76 78 80
|
3eqtr3d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 𝑘 ) = ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) |
| 82 |
81
|
fveq2d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 1 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) ) |
| 83 |
74 82
|
eqtrd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) ) |
| 84 |
63 83
|
eqeq12d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) = ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) ↔ ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) ) ) |
| 85 |
68 84
|
syl5bb |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ↔ ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) ) ) |
| 86 |
|
wkslem1 |
⊢ ( 𝑥 = ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) → ( if- ( ( 𝑃 ‘ 𝑥 ) = ( 𝑃 ‘ ( 𝑥 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑥 ) ) = { ( 𝑃 ‘ 𝑥 ) } , { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑥 ) ) ) ↔ if- ( ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) } , { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) ) ) |
| 87 |
5 1
|
wlkprop |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑥 ) = ( 𝑃 ‘ ( 𝑥 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑥 ) ) = { ( 𝑃 ‘ 𝑥 ) } , { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
| 88 |
87
|
simp3d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ∀ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑥 ) = ( 𝑃 ‘ ( 𝑥 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑥 ) ) = { ( 𝑃 ‘ 𝑥 ) } , { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) |
| 89 |
88
|
adantr |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ∀ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑥 ) = ( 𝑃 ‘ ( 𝑥 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑥 ) ) = { ( 𝑃 ‘ 𝑥 ) } , { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) |
| 90 |
18
|
nn0cnd |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℂ ) |
| 91 |
90
|
adantr |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ♯ ‘ 𝐹 ) ∈ ℂ ) |
| 92 |
91 51 52
|
sub32d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝐹 ) − 𝑘 ) − 1 ) = ( ( ( ♯ ‘ 𝐹 ) − 1 ) − 𝑘 ) ) |
| 93 |
37
|
adantr |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ♯ ‘ 𝐹 ) − 1 ) = ( ( ♯ ‘ 𝑃 ) − 2 ) ) |
| 94 |
93
|
oveq1d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝐹 ) − 1 ) − 𝑘 ) = ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) |
| 95 |
92 94
|
eqtrd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝐹 ) − 𝑘 ) − 1 ) = ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) |
| 96 |
|
ubmelm1fzo |
⊢ ( 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ( ( ♯ ‘ 𝐹 ) − 𝑘 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 97 |
96
|
adantl |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝐹 ) − 𝑘 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 98 |
95 97
|
eqeltrrd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
| 99 |
86 89 98
|
rspcdva |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → if- ( ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) } , { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) ) |
| 100 |
|
dfifp2 |
⊢ ( if- ( ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) } , { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) ↔ ( ( ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) } ) ∧ ( ¬ ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) → { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) ) ) |
| 101 |
99 100
|
sylib |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) } ) ∧ ( ¬ ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) → { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) ) ) |
| 102 |
101
|
simpld |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) } ) ) |
| 103 |
85 102
|
sylbid |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) } ) ) |
| 104 |
103
|
imp |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) } ) |
| 105 |
65 67 104
|
3eqtr4rd |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) = { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } ) |
| 106 |
42 105
|
eqtrd |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) = { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } ) |
| 107 |
85
|
notbid |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ¬ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ↔ ¬ ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) ) ) |
| 108 |
101
|
simprd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ¬ ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) = ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) → { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) ) |
| 109 |
107 108
|
sylbid |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ¬ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) → { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) ) |
| 110 |
109
|
imp |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ) → { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) |
| 111 |
|
prcom |
⊢ { ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) , ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } = { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) , ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } |
| 112 |
63 83
|
preq12d |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → { ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) , ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ) |
| 113 |
111 112
|
eqtr3id |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) , ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ) |
| 114 |
113
|
adantr |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ) → { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) , ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } = { ( 𝑃 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) , ( 𝑃 ‘ ( ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) + 1 ) ) } ) |
| 115 |
41
|
adantr |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ ( ( ( ♯ ‘ 𝑃 ) − 2 ) − 𝑘 ) ) ) ) |
| 116 |
110 114 115
|
3sstr4d |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) ) → { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) , ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) ) |
| 117 |
106 116
|
ifpimpda |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → if- ( ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) = { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } , { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) , ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) ) ) |
| 118 |
28 117
|
syldan |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) ) → if- ( ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) = { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } , { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) , ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) ) ) |
| 119 |
118
|
ralrimiva |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) if- ( ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) = { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } , { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) , ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) ) ) |
| 120 |
|
wlkv |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) |
| 121 |
120
|
simp1d |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝐺 ∈ V ) |
| 122 |
5 1
|
iswlkg |
⊢ ( 𝐺 ∈ V → ( ( reverse ‘ 𝐹 ) ( Walks ‘ 𝐺 ) ( reverse ‘ 𝑃 ) ↔ ( ( reverse ‘ 𝐹 ) ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ ( reverse ‘ 𝑃 ) : ( 0 ... ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) if- ( ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) = { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } , { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) , ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) ) ) ) ) |
| 123 |
121 122
|
syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( reverse ‘ 𝐹 ) ( Walks ‘ 𝐺 ) ( reverse ‘ 𝑃 ) ↔ ( ( reverse ‘ 𝐹 ) ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ ( reverse ‘ 𝑃 ) : ( 0 ... ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ ( reverse ‘ 𝐹 ) ) ) if- ( ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) = ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) = { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) } , { ( ( reverse ‘ 𝑃 ) ‘ 𝑘 ) , ( ( reverse ‘ 𝑃 ) ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( ( reverse ‘ 𝐹 ) ‘ 𝑘 ) ) ) ) ) ) |
| 124 |
4 25 119 123
|
mpbir3and |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( reverse ‘ 𝐹 ) ( Walks ‘ 𝐺 ) ( reverse ‘ 𝑃 ) ) |