Theorem clwwlknlbonbgr1

Description: The last but one vertex in a closed walk is a neighbor of the first vertex of the closed walk. (Contributed by AV, 17-Feb-2022)

	Ref	Expression
Assertion	clwwlknlbonbgr1	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ) → ( 𝑊 ‘ ( 𝑁 − 1 ) ) ∈ ( 𝐺 NeighbVtx ( 𝑊 ‘ 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
3	1 2	clwwlknp	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
4		lsw	⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) )
5		fvoveq1	⊢ ( ( ♯ ‘ 𝑊 ) = 𝑁 → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 ‘ ( 𝑁 − 1 ) ) )
6	4 5	sylan9eq	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( 𝑁 − 1 ) ) )
7	6	preq1d	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) → { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } = { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 0 ) } )
8	7	eleq1d	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ↔ { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
9	8	biimpd	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
10	9	a1d	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) )
11	10	3imp	⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) )
12	3 11	syl	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) )
13	12	adantl	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ) → { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) )
14	2	nbusgreledg	⊢ ( 𝐺 ∈ USGraph → ( ( 𝑊 ‘ ( 𝑁 − 1 ) ) ∈ ( 𝐺 NeighbVtx ( 𝑊 ‘ 0 ) ) ↔ { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
15	14	adantr	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ) → ( ( 𝑊 ‘ ( 𝑁 − 1 ) ) ∈ ( 𝐺 NeighbVtx ( 𝑊 ‘ 0 ) ) ↔ { ( 𝑊 ‘ ( 𝑁 − 1 ) ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
16	13 15	mpbird	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ) → ( 𝑊 ‘ ( 𝑁 − 1 ) ) ∈ ( 𝐺 NeighbVtx ( 𝑊 ‘ 0 ) ) )