Description: The order of a star graph S_N. (Contributed by AV, 12-Sep-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | stgrvtx0.g | ⊢ 𝐺 = ( StarGr ‘ 𝑁 ) | |
| stgrvtx0.v | ⊢ 𝑉 = ( Vtx ‘ 𝐺 ) | ||
| Assertion | stgrorder | ⊢ ( 𝑁 ∈ ℕ0 → ( ♯ ‘ 𝑉 ) = ( 𝑁 + 1 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | stgrvtx0.g | ⊢ 𝐺 = ( StarGr ‘ 𝑁 ) | |
| 2 | stgrvtx0.v | ⊢ 𝑉 = ( Vtx ‘ 𝐺 ) | |
| 3 | 1 | fveq2i | ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) |
| 4 | 2 3 | eqtri | ⊢ 𝑉 = ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) |
| 5 | stgrvtx | ⊢ ( 𝑁 ∈ ℕ0 → ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) ) | |
| 6 | 4 5 | eqtrid | ⊢ ( 𝑁 ∈ ℕ0 → 𝑉 = ( 0 ... 𝑁 ) ) |
| 7 | 6 | fveq2d | ⊢ ( 𝑁 ∈ ℕ0 → ( ♯ ‘ 𝑉 ) = ( ♯ ‘ ( 0 ... 𝑁 ) ) ) |
| 8 | hashfz0 | ⊢ ( 𝑁 ∈ ℕ0 → ( ♯ ‘ ( 0 ... 𝑁 ) ) = ( 𝑁 + 1 ) ) | |
| 9 | 7 8 | eqtrd | ⊢ ( 𝑁 ∈ ℕ0 → ( ♯ ‘ 𝑉 ) = ( 𝑁 + 1 ) ) |