Theorem hashclwwlkn0

Description: The number of closed walks (defined as words) with a fixed length is the sum of the sizes of all equivalence classes according to .~ . (Contributed by Alexander van der Vekens, 10-Apr-2018) (Revised by AV, 30-Apr-2021)

	Ref	Expression
Hypotheses	erclwwlkn.w	$$⊢W=N\mathrm{ClWWalksN}G$$
	erclwwlkn.r	$$⊢\underset{\dot{}}{\sim }=\left\{⟨t,u⟩|\left(t\in W\wedge u\in W\wedge \exists n\in \left(0\dots N\right)t=u\mathrm{cyclShift}n\right)\right\}$$
Assertion	hashclwwlkn0	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{Fin}\to \left|W\right|=\sum _{x\in W/\underset{\dot{}}{\sim }}\left|x\right|$$

Proof

Step	Hyp	Ref	Expression
1		erclwwlkn.w	$$⊢W=N\mathrm{ClWWalksN}G$$
2		erclwwlkn.r	$$⊢\underset{\dot{}}{\sim }=\left\{⟨t,u⟩|\left(t\in W\wedge u\in W\wedge \exists n\in \left(0\dots N\right)t=u\mathrm{cyclShift}n\right)\right\}$$
3	1 2	erclwwlkn	$$⊢\underset{\dot{}}{\sim }\mathrm{Er}W$$
4	3	a1i	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{Fin}\to \underset{\dot{}}{\sim }\mathrm{Er}W$$
5		clwwlknfi	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{Fin}\to N\mathrm{ClWWalksN}G\in \mathrm{Fin}$$
6	1 5	eqeltrid	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{Fin}\to W\in \mathrm{Fin}$$
7	4 6	qshash	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{Fin}\to \left|W\right|=\sum _{x\in W/\underset{\dot{}}{\sim }}\left|x\right|$$