Description: The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018) (Revised by AV, 1-May-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | wlklenvp1 | ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wlkcl | ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) | |
| 2 | eqid | ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) | |
| 3 | 2 | wlkp | ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) |
| 4 | ffz0hash | ⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) | |
| 5 | 1 3 4 | syl2anc | ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) |