| Step |
Hyp |
Ref |
Expression |
| 1 |
|
wlkv |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) |
| 2 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
| 3 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
| 4 |
2 3
|
iswlk |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
| 5 |
|
simpr1 |
⊢ ( ( ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) → 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) |
| 6 |
|
simpr2 |
⊢ ( ( ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) → 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) |
| 7 |
|
df-ifp |
⊢ ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) ∨ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 8 |
|
dfsn2 |
⊢ { ( 𝑃 ‘ 𝑘 ) } = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ 𝑘 ) } |
| 9 |
|
preq2 |
⊢ ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ 𝑘 ) } = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) |
| 10 |
8 9
|
syl5eq |
⊢ ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → { ( 𝑃 ‘ 𝑘 ) } = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) |
| 11 |
10
|
eqeq2d |
⊢ ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ↔ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 12 |
11
|
biimpa |
⊢ ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) |
| 13 |
12
|
a1d |
⊢ ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) → ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 14 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) → 𝐺 ∈ UPGraph ) |
| 15 |
|
simpl |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) |
| 16 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
| 17 |
3 16
|
upgredginwlk |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) → ( 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ) |
| 18 |
14 15 17
|
syl2anr |
⊢ ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) → ( 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ) |
| 19 |
18
|
imp |
⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) |
| 20 |
|
simprr |
⊢ ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) → 𝐺 ∈ UPGraph ) |
| 21 |
20
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝐺 ∈ UPGraph ) |
| 22 |
21
|
adantr |
⊢ ( ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) → 𝐺 ∈ UPGraph ) |
| 23 |
22
|
adantr |
⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → 𝐺 ∈ UPGraph ) |
| 24 |
|
simplr |
⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) |
| 25 |
|
simprr |
⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
| 26 |
|
df-ne |
⊢ ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ↔ ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) |
| 27 |
|
fvexd |
⊢ ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( 𝑃 ‘ 𝑘 ) ∈ V ) |
| 28 |
|
fvexd |
⊢ ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ) |
| 29 |
|
id |
⊢ ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) |
| 30 |
27 28 29
|
3jca |
⊢ ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( ( 𝑃 ‘ 𝑘 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) |
| 31 |
26 30
|
sylbir |
⊢ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( ( 𝑃 ‘ 𝑘 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) |
| 32 |
31
|
adantr |
⊢ ( ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 𝑃 ‘ 𝑘 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) |
| 33 |
32
|
adantl |
⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( 𝑃 ‘ 𝑘 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) |
| 34 |
2 16
|
upgredgpr |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ∧ ( ( 𝑃 ‘ 𝑘 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
| 35 |
23 24 25 33 34
|
syl31anc |
⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
| 36 |
35
|
eqcomd |
⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) |
| 37 |
36
|
exp31 |
⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) → ( ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) |
| 38 |
19 37
|
mpd |
⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 39 |
38
|
com12 |
⊢ ( ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 40 |
13 39
|
jaoi |
⊢ ( ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) ∨ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 41 |
40
|
com12 |
⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) ∨ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 42 |
7 41
|
syl5bi |
⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 43 |
42
|
ralimdva |
⊢ ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 44 |
43
|
ex |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) |
| 45 |
44
|
com23 |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) |
| 46 |
45
|
3impia |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 47 |
46
|
impcom |
⊢ ( ( ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) |
| 48 |
5 6 47
|
3jca |
⊢ ( ( ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ 𝐺 ∈ UPGraph ) ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 49 |
48
|
exp31 |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐺 ∈ UPGraph → ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) ) |
| 50 |
49
|
com23 |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐺 ∈ UPGraph → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) ) |
| 51 |
4 50
|
sylbid |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ UPGraph → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) ) |
| 52 |
51
|
impd |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐺 ∈ UPGraph ) → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) |
| 53 |
52
|
impcom |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐺 ∈ UPGraph ) ∧ ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
| 54 |
2 3
|
isupwlk |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ( UPWalks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) |
| 55 |
54
|
adantl |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐺 ∈ UPGraph ) ∧ ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) → ( 𝐹 ( UPWalks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) |
| 56 |
53 55
|
mpbird |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐺 ∈ UPGraph ) ∧ ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) → 𝐹 ( UPWalks ‘ 𝐺 ) 𝑃 ) |
| 57 |
56
|
exp31 |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ UPGraph → ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → 𝐹 ( UPWalks ‘ 𝐺 ) 𝑃 ) ) ) |
| 58 |
1 57
|
mpid |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ UPGraph → 𝐹 ( UPWalks ‘ 𝐺 ) 𝑃 ) ) |
| 59 |
58
|
impcom |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) → 𝐹 ( UPWalks ‘ 𝐺 ) 𝑃 ) |