Theorem stgrorder

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 ) )

Proof

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 ) )