Theorem clwwlknbp

Description: Basic properties of a closed walk of a fixed length as word. (Contributed by AV, 30-Apr-2021) (Proof shortened by AV, 22-Mar-2022)

	Ref	Expression
Hypothesis	clwwlknwrd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	clwwlknbp	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		clwwlknwrd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	clwwlknwrd	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → 𝑊 ∈ Word 𝑉 )
3		clwwlknlen	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( ♯ ‘ 𝑊 ) = 𝑁 )
4	2 3	jca	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) )