Step |
Hyp |
Ref |
Expression |
1 |
|
wlkonl1iedg.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
2 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
3 |
2
|
wlkonprop |
⊢ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ) |
4 |
|
fveq2 |
⊢ ( 𝑘 = 0 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 0 ) ) |
5 |
|
fv0p1e1 |
⊢ ( 𝑘 = 0 → ( 𝑃 ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ 1 ) ) |
6 |
4 5
|
preq12d |
⊢ ( 𝑘 = 0 → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ) |
7 |
6
|
sseq1d |
⊢ ( 𝑘 = 0 → ( { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 ↔ { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ 𝑒 ) ) |
8 |
7
|
rexbidv |
⊢ ( 𝑘 = 0 → ( ∃ 𝑒 ∈ ran 𝐼 { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ ran 𝐼 { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ 𝑒 ) ) |
9 |
1
|
wlkvtxiedg |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∃ 𝑒 ∈ ran 𝐼 { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 ) |
10 |
9
|
adantr |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∃ 𝑒 ∈ ran 𝐼 { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) ∧ ( ♯ ‘ 𝐹 ) ≠ 0 ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∃ 𝑒 ∈ ran 𝐼 { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 ) |
12 |
|
wlkcl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
13 |
|
elnnne0 |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ ↔ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝐹 ) ≠ 0 ) ) |
14 |
13
|
simplbi2 |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( ( ♯ ‘ 𝐹 ) ≠ 0 → ( ♯ ‘ 𝐹 ) ∈ ℕ ) ) |
15 |
|
lbfzo0 |
⊢ ( 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ ( ♯ ‘ 𝐹 ) ∈ ℕ ) |
16 |
14 15
|
syl6ibr |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( ( ♯ ‘ 𝐹 ) ≠ 0 → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
17 |
12 16
|
syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( ♯ ‘ 𝐹 ) ≠ 0 → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
18 |
17
|
adantr |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) → ( ( ♯ ‘ 𝐹 ) ≠ 0 → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
19 |
18
|
imp |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) ∧ ( ♯ ‘ 𝐹 ) ≠ 0 ) → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
20 |
8 11 19
|
rspcdva |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) ∧ ( ♯ ‘ 𝐹 ) ≠ 0 ) → ∃ 𝑒 ∈ ran 𝐼 { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ 𝑒 ) |
21 |
|
fvex |
⊢ ( 𝑃 ‘ 0 ) ∈ V |
22 |
|
fvex |
⊢ ( 𝑃 ‘ 1 ) ∈ V |
23 |
21 22
|
prss |
⊢ ( ( ( 𝑃 ‘ 0 ) ∈ 𝑒 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑒 ) ↔ { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ 𝑒 ) |
24 |
|
eleq1 |
⊢ ( ( 𝑃 ‘ 0 ) = 𝐴 → ( ( 𝑃 ‘ 0 ) ∈ 𝑒 ↔ 𝐴 ∈ 𝑒 ) ) |
25 |
|
ax-1 |
⊢ ( 𝐴 ∈ 𝑒 → ( ( 𝑃 ‘ 1 ) ∈ 𝑒 → 𝐴 ∈ 𝑒 ) ) |
26 |
24 25
|
syl6bi |
⊢ ( ( 𝑃 ‘ 0 ) = 𝐴 → ( ( 𝑃 ‘ 0 ) ∈ 𝑒 → ( ( 𝑃 ‘ 1 ) ∈ 𝑒 → 𝐴 ∈ 𝑒 ) ) ) |
27 |
26
|
adantl |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) → ( ( 𝑃 ‘ 0 ) ∈ 𝑒 → ( ( 𝑃 ‘ 1 ) ∈ 𝑒 → 𝐴 ∈ 𝑒 ) ) ) |
28 |
27
|
impd |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) → ( ( ( 𝑃 ‘ 0 ) ∈ 𝑒 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑒 ) → 𝐴 ∈ 𝑒 ) ) |
29 |
23 28
|
syl5bir |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) → ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ 𝑒 → 𝐴 ∈ 𝑒 ) ) |
30 |
29
|
reximdv |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) → ( ∃ 𝑒 ∈ ran 𝐼 { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ 𝑒 → ∃ 𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒 ) ) |
31 |
30
|
adantr |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) ∧ ( ♯ ‘ 𝐹 ) ≠ 0 ) → ( ∃ 𝑒 ∈ ran 𝐼 { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ 𝑒 → ∃ 𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒 ) ) |
32 |
20 31
|
mpd |
⊢ ( ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) ∧ ( ♯ ‘ 𝐹 ) ≠ 0 ) → ∃ 𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒 ) |
33 |
32
|
ex |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) → ( ( ♯ ‘ 𝐹 ) ≠ 0 → ∃ 𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒 ) ) |
34 |
33
|
3adant3 |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( ( ♯ ‘ 𝐹 ) ≠ 0 → ∃ 𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒 ) ) |
35 |
34
|
3ad2ant3 |
⊢ ( ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( ♯ ‘ 𝐹 ) ≠ 0 → ∃ 𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒 ) ) |
36 |
3 35
|
syl |
⊢ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → ( ( ♯ ‘ 𝐹 ) ≠ 0 → ∃ 𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒 ) ) |
37 |
36
|
imp |
⊢ ( ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 ∧ ( ♯ ‘ 𝐹 ) ≠ 0 ) → ∃ 𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒 ) |