Theorem clwlksndivn

Description: The size of the set of closed walks of prime length N is divisible by N . This corresponds to statement 9 in Huneke p. 2: "It follows that, if p is a prime number, then the number of closed walks of length p is divisible by p". (Contributed by Alexander van der Vekens, 6-Jul-2018) (Revised by AV, 4-May-2021)

	Ref	Expression
Assertion	clwlksndivn	|- ( ( G e. FinUSGraph /\ N e. Prime ) -> N || ( # ` { c e. ( ClWalks ` G ) | ( # ` ( 1st ` c ) ) = N } ) )

Proof

Step	Hyp	Ref	Expression
1		clwwlkndivn	|- ( ( G e. FinUSGraph /\ N e. Prime ) -> N || ( # ` ( N ClWWalksN G ) ) )
2		fusgrusgr	|- ( G e. FinUSGraph -> G e. USGraph )
3		usgruspgr	|- ( G e. USGraph -> G e. USPGraph )
4	2 3	syl	|- ( G e. FinUSGraph -> G e. USPGraph )
5		prmnn	|- ( N e. Prime -> N e. NN )
6		clwlkssizeeq	|- ( ( G e. USPGraph /\ N e. NN ) -> ( # ` ( N ClWWalksN G ) ) = ( # ` { c e. ( ClWalks ` G ) | ( # ` ( 1st ` c ) ) = N } ) )
7	4 5 6	syl2an	|- ( ( G e. FinUSGraph /\ N e. Prime ) -> ( # ` ( N ClWWalksN G ) ) = ( # ` { c e. ( ClWalks ` G ) | ( # ` ( 1st ` c ) ) = N } ) )
8	1 7	breqtrd	|- ( ( G e. FinUSGraph /\ N e. Prime ) -> N || ( # ` { c e. ( ClWalks ` G ) | ( # ` ( 1st ` c ) ) = N } ) )