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 ∈ ℙ ) → 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 ∈ ℙ ) → ( 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 ) ) } ) ) ∈ ℕ0 )
8	3 6 7	3syl	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → ( ♯ ‘ ( ( 𝑁 ClWWalksN 𝐺 ) / { ⟨ 𝑡 , 𝑢 ⟩ ∣ ( 𝑡 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ 𝑢 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) } ) ) ∈ ℕ0 )
9	8	nn0zd	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → ( ♯ ‘ ( ( 𝑁 ClWWalksN 𝐺 ) / { ⟨ 𝑡 , 𝑢 ⟩ ∣ ( 𝑡 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ 𝑢 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) } ) ) ∈ ℤ )
10		prmz	⊢ ( 𝑁 ∈ ℙ → 𝑁 ∈ ℤ )
11	10	adantl	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → 𝑁 ∈ ℤ )
12		dvdsmul2	⊢ ( ( ( ♯ ‘ ( ( 𝑁 ClWWalksN 𝐺 ) / { ⟨ 𝑡 , 𝑢 ⟩ ∣ ( 𝑡 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ 𝑢 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) } ) ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → 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 ∈ ℙ ) → 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 ∈ ℙ ) → ( ♯ ‘ ( 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 ∈ ℙ ) → 𝑁 ∥ ( ♯ ‘ ( 𝑁 ClWWalksN 𝐺 ) ) )