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Metamath Proof Explorer

Theorem clwlkiswlk

Description: A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018) (Revised by AV, 16-Feb-2021) (Proof shortened by AV, 30-Oct-2021)

		Ref	Expression
	Assertion	clwlkiswlk	⊢ ( 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		isclwlk	⊢ ( 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
2	1	simplbi	⊢ ( 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )