Metamath Proof Explorer


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 FinUSGraph N N c ClWalks G | 1 st c = N

Proof

Step Hyp Ref Expression
1 clwwlkndivn G FinUSGraph N N N ClWWalksN G
2 fusgrusgr G FinUSGraph G USGraph
3 usgruspgr G USGraph G USHGraph
4 2 3 syl G FinUSGraph G USHGraph
5 prmnn N N
6 clwlkssizeeq G USHGraph N N ClWWalksN G = c ClWalks G | 1 st c = N
7 4 5 6 syl2an G FinUSGraph N N ClWWalksN G = c ClWalks G | 1 st c = N
8 1 7 breqtrd G FinUSGraph N N c ClWalks G | 1 st c = N