Theorem erclwwlknrel

Description: .~ is a relation. (Contributed by Alexander van der Vekens, 25-Mar-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	erclwwlknrel	$$⊢\mathrm{Rel}\underset{\dot{}}{\sim }$$

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	2	relopabi	$$⊢\mathrm{Rel}\underset{\dot{}}{\sim }$$