Step |
Hyp |
Ref |
Expression |
1 |
|
wwlknbp1 |
⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) |
2 |
|
simp2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) |
3 |
|
nn0z |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ ) |
4 |
|
fzval3 |
⊢ ( 𝑁 ∈ ℤ → ( 0 ... 𝑁 ) = ( 0 ..^ ( 𝑁 + 1 ) ) ) |
5 |
3 4
|
syl |
⊢ ( 𝑁 ∈ ℕ0 → ( 0 ... 𝑁 ) = ( 0 ..^ ( 𝑁 + 1 ) ) ) |
6 |
5
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → ( 0 ... 𝑁 ) = ( 0 ..^ ( 𝑁 + 1 ) ) ) |
7 |
6
|
eleq2d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → ( 𝑖 ∈ ( 0 ... 𝑁 ) ↔ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) ) ) |
8 |
7
|
biimpa |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ∧ 𝑖 ∈ ( 0 ... 𝑁 ) ) → 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) ) |
9 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ..^ ( 𝑁 + 1 ) ) ) |
10 |
9
|
eleq2d |
⊢ ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) → ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ↔ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) ) ) |
11 |
10
|
3ad2ant3 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ↔ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) ) ) |
12 |
11
|
adantr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ∧ 𝑖 ∈ ( 0 ... 𝑁 ) ) → ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ↔ 𝑖 ∈ ( 0 ..^ ( 𝑁 + 1 ) ) ) ) |
13 |
8 12
|
mpbird |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ∧ 𝑖 ∈ ( 0 ... 𝑁 ) ) → 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
14 |
|
wrdsymbcl |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ 𝑖 ) ∈ ( Vtx ‘ 𝐺 ) ) |
15 |
2 13 14
|
syl2an2r |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ∧ 𝑖 ∈ ( 0 ... 𝑁 ) ) → ( 𝑊 ‘ 𝑖 ) ∈ ( Vtx ‘ 𝐺 ) ) |
16 |
15
|
ralrimiva |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ( 𝑊 ‘ 𝑖 ) ∈ ( Vtx ‘ 𝐺 ) ) |
17 |
1 16
|
syl |
⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ∀ 𝑖 ∈ ( 0 ... 𝑁 ) ( 𝑊 ‘ 𝑖 ) ∈ ( Vtx ‘ 𝐺 ) ) |