Metamath Proof Explorer


Theorem clwwlkndivn

Description: The size of the set of closed walks (defined as words) of length N is divisible by N if N is a prime number. (Contributed by Alexander van der Vekens, 17-Jun-2018) (Revised by AV, 2-May-2021)

Ref Expression
Assertion clwwlkndivn G FinUSGraph N N N ClWWalksN G

Proof

Step Hyp Ref Expression
1 eqid Vtx G = Vtx G
2 1 fusgrvtxfi G FinUSGraph Vtx G Fin
3 2 adantr G FinUSGraph N Vtx G Fin
4 eqid N ClWWalksN G = N ClWWalksN G
5 eqid t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n = t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n
6 4 5 qerclwwlknfi Vtx G Fin N ClWWalksN G / t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n Fin
7 hashcl N ClWWalksN G / t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n Fin N ClWWalksN G / t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n 0
8 3 6 7 3syl G FinUSGraph N N ClWWalksN G / t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n 0
9 8 nn0zd G FinUSGraph N N ClWWalksN G / t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n
10 prmz N N
11 10 adantl G FinUSGraph N N
12 dvdsmul2 N ClWWalksN G / t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n N N N ClWWalksN G / t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n N
13 9 11 12 syl2anc G FinUSGraph N N N ClWWalksN G / t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n N
14 4 5 fusgrhashclwwlkn G FinUSGraph N N ClWWalksN G = N ClWWalksN G / t u | t N ClWWalksN G u N ClWWalksN G n 0 N t = u cyclShift n N
15 13 14 breqtrrd G FinUSGraph N N N ClWWalksN G