Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
2 |
1
|
wwlknbp |
⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) ) |
3 |
|
iswwlksn |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) ) |
4 |
3
|
3ad2ant2 |
⊢ ( ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) ) |
5 |
|
simpl |
⊢ ( ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → 𝑊 ∈ ( WWalks ‘ 𝐺 ) ) |
6 |
4 5
|
syl6bi |
⊢ ( ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( WWalks ‘ 𝐺 ) ) ) |
7 |
2 6
|
mpcom |
⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( WWalks ‘ 𝐺 ) ) |