Step |
Hyp |
Ref |
Expression |
1 |
|
trlsegvdeg.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
trlsegvdeg.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
trlsegvdeg.f |
⊢ ( 𝜑 → Fun 𝐼 ) |
4 |
|
trlsegvdeg.n |
⊢ ( 𝜑 → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
5 |
|
trlsegvdeg.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
6 |
|
trlsegvdeg.w |
⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) |
7 |
|
trliswlk |
⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |
8 |
1
|
wlkpvtx |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) → ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 ) ) |
9 |
|
elfzofz |
⊢ ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ) |
10 |
8 9
|
impel |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 ) |
11 |
1
|
wlkpvtx |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( 𝑁 + 1 ) ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) → ( 𝑃 ‘ ( 𝑁 + 1 ) ) ∈ 𝑉 ) ) |
12 |
|
fzofzp1 |
⊢ ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝑁 + 1 ) ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ) |
13 |
11 12
|
impel |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ ( 𝑁 + 1 ) ) ∈ 𝑉 ) |
14 |
10 13
|
jca |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 ∧ ( 𝑃 ‘ ( 𝑁 + 1 ) ) ∈ 𝑉 ) ) |
15 |
14
|
ex |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 ∧ ( 𝑃 ‘ ( 𝑁 + 1 ) ) ∈ 𝑉 ) ) ) |
16 |
6 7 15
|
3syl |
⊢ ( 𝜑 → ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 ∧ ( 𝑃 ‘ ( 𝑁 + 1 ) ) ∈ 𝑉 ) ) ) |
17 |
4 16
|
mpd |
⊢ ( 𝜑 → ( ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 ∧ ( 𝑃 ‘ ( 𝑁 + 1 ) ) ∈ 𝑉 ) ) |