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 e. NN0 -> ( # ` 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 e. NN0 -> ( Vtx ` ( StarGr ` N ) ) = ( 0 ... N ) )
6	4 5	eqtrid	|- ( N e. NN0 -> V = ( 0 ... N ) )
7	6	fveq2d	|- ( N e. NN0 -> ( # ` V ) = ( # ` ( 0 ... N ) ) )
8		hashfz0	|- ( N e. NN0 -> ( # ` ( 0 ... N ) ) = ( N + 1 ) )
9	7 8	eqtrd	|- ( N e. NN0 -> ( # ` V ) = ( N + 1 ) )