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Metamath Proof Explorer

Theorem clwwlkclwwlkn

Description: A closed walk of a fixed length as word is a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018) (Revised by AV, 24-Apr-2021) (Proof shortened by AV, 22-Mar-2022)

		Ref	Expression
	Assertion	clwwlkclwwlkn	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		isclwwlkn	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) )
2	1	simplbi	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) )