Step |
Hyp |
Ref |
Expression |
1 |
|
wwlknbp1 |
⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) |
2 |
|
lsw |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
3 |
2
|
3ad2ant2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
4 |
|
oveq1 |
⊢ ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) = ( ( 𝑁 + 1 ) − 1 ) ) |
5 |
4
|
3ad2ant3 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → ( ( ♯ ‘ 𝑊 ) − 1 ) = ( ( 𝑁 + 1 ) − 1 ) ) |
6 |
|
nn0cn |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ ) |
7 |
|
pncan1 |
⊢ ( 𝑁 ∈ ℂ → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 ) |
8 |
6 7
|
syl |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 ) |
9 |
8
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 ) |
10 |
5 9
|
eqtrd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝑁 ) |
11 |
10
|
fveq2d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 ‘ 𝑁 ) ) |
12 |
3 11
|
eqtr2d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → ( 𝑊 ‘ 𝑁 ) = ( lastS ‘ 𝑊 ) ) |
13 |
1 12
|
syl |
⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝑊 ‘ 𝑁 ) = ( lastS ‘ 𝑊 ) ) |