Theorem clwwlkbp

Description: Basic properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018) (Revised by AV, 24-Apr-2021)

	Ref	Expression
Hypothesis	clwwlkbp.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	clwwlkbp	⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		clwwlkbp.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		elfvex	⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → 𝐺 ∈ V )
3		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
4	1 3	isclwwlk	⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ↔ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
5	4	simp1bi	⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) )
6		3anass	⊢ ( ( 𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) ↔ ( 𝐺 ∈ V ∧ ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) ) )
7	2 5 6	sylanbrc	⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) )