Theorem erclwwlkn

Description: .~ is an equivalence relation over the set of closed walks (defined as words) with a fixed length. (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	erclwwlkn	$$⊢\underset{\dot{}}{\sim }\mathrm{Er}W$$

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	erclwwlknrel	$$⊢\mathrm{Rel}\underset{\dot{}}{\sim }$$
4	1 2	erclwwlknsym	$$⊢x\underset{\dot{}}{\sim }y\to y\underset{\dot{}}{\sim }x$$
5	1 2	erclwwlkntr	$$⊢\left(x\underset{\dot{}}{\sim }y\wedge y\underset{\dot{}}{\sim }z\right)\to x\underset{\dot{}}{\sim }z$$
6	1 2	erclwwlknref	$$⊢x\in W\leftrightarrow x\underset{\dot{}}{\sim }x$$
7	3 4 5 6	iseri	$$⊢\underset{\dot{}}{\sim }\mathrm{Er}W$$