Step |
Hyp |
Ref |
Expression |
1 |
|
2pthnloop.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
2 |
|
pthiswlk |
⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |
3 |
|
wlkv |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) |
4 |
2 3
|
syl |
⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) |
5 |
|
ispth |
⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ) |
6 |
|
istrl |
⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) ) |
7 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
8 |
7 1
|
iswlkg |
⊢ ( 𝐺 ∈ V → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) |
9 |
8
|
anbi1d |
⊢ ( 𝐺 ∈ V → ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) ↔ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∧ Fun ◡ 𝐹 ) ) ) |
10 |
6 9
|
syl5bb |
⊢ ( 𝐺 ∈ V → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∧ Fun ◡ 𝐹 ) ) ) |
11 |
|
pthdadjvtx |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝑖 ) ≠ ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) |
12 |
11
|
ad5ant245 |
⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝑖 ) ≠ ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) |
13 |
12
|
neneqd |
⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) |
14 |
|
ifpfal |
⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ↔ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) |
15 |
14
|
adantl |
⊢ ( ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ↔ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) |
16 |
|
fvexd |
⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( 𝑃 ‘ 𝑖 ) ∈ V ) |
17 |
|
fvexd |
⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( 𝑃 ‘ ( 𝑖 + 1 ) ) ∈ V ) |
18 |
|
neqne |
⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( 𝑃 ‘ 𝑖 ) ≠ ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) |
19 |
|
fvexd |
⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ V ) |
20 |
|
prsshashgt1 |
⊢ ( ( ( ( 𝑃 ‘ 𝑖 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑖 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑖 ) ≠ ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ V ) → ( { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) → 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) |
21 |
16 17 18 19 20
|
syl31anc |
⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) → 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) |
22 |
21
|
adantl |
⊢ ( ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) → ( { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) → 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) |
23 |
15 22
|
sylbid |
⊢ ( ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) |
24 |
13 23
|
mpdan |
⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) |
25 |
24
|
ralimdva |
⊢ ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) |
26 |
25
|
ex |
⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) → ( 1 < ( ♯ ‘ 𝐹 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) |
27 |
26
|
com23 |
⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) |
28 |
27
|
exp31 |
⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) |
29 |
28
|
com24 |
⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ( ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) |
30 |
29
|
3impia |
⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) → ( ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) |
31 |
30
|
exp4c |
⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) → ( Fun ◡ 𝐹 → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) ) |
32 |
31
|
imp |
⊢ ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∧ Fun ◡ 𝐹 ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) |
33 |
10 32
|
syl6bi |
⊢ ( 𝐺 ∈ V → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) ) |
34 |
33
|
com24 |
⊢ ( 𝐺 ∈ V → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) ) |
35 |
34
|
com14 |
⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( 𝐺 ∈ V → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) ) |
36 |
35
|
3imp |
⊢ ( ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝐺 ∈ V → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) |
37 |
36
|
com12 |
⊢ ( 𝐺 ∈ V → ( ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) |
38 |
5 37
|
syl5bi |
⊢ ( 𝐺 ∈ V → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) |
39 |
38
|
3ad2ant1 |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) |
40 |
4 39
|
mpcom |
⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) |
41 |
40
|
pm2.43i |
⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) |
42 |
41
|
imp |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) |